It is clear from the main partitions page that the approximation function of Hardy and Ramanujan $$ pp(n):= \frac{1}{4n\sqrt{3}}\exp\left(\pi \sqrt{2n/3}\right) $$ is not accurate enough to obtain exact values for \(p(n)\) for large values of \(n\). Rademacher later derived an infinite series that converges very rapidly to \(p(n)\), and in fact converges so rapidly that it can be used effectively to compute the exact value of \(p(n)\) for large \(n\). This, and similar series for other partition functions, are known as Hardy–Ramanujan-Rademacher-type series . To describe this formula, for each positive integers \(h\) and \(k\) with \(1\leq h \leq k\), let $$ s(h,k):=\sum _{r=1}^{k-1} \frac{r}{k} \left(-\left\lfloor \frac{h r}{k}\right\rfloor +\frac{h r}{k}-\frac{1}{2}\right), $$ let $$ A_k(n):=\sum _{\stackrel{h=1}{\gcd(h,k)=1}}^{k-1} \exp \left(i \pi s(h,k)-2 i \pi h n/k\right ), $$ and then \begin{align*} p(n) &= \frac{1}{\pi \sqrt{2}}\sum_{k=1}^{\infty}\sqrt{k}A_k(n)\frac{d \sinh \left( \frac{\pi}{k} \sqrt{\frac{2}{3}( n-1/24)} \right)/\sqrt{n-1/24} }{d\,n}\\ &= \frac{1}{\pi \sqrt{2}}\sum_{k=1}^{\infty}\sqrt{k}A_k(n) \left( \frac{\pi \cosh \left( \frac{\pi}{k} \sqrt{\frac{2}{3}( n-1/24)} \right)}{\sqrt{6} k \left(n-1/24\right)}-\frac{\sinh \left( \frac{\pi}{k} \sqrt{\frac{2}{3}( n-1/24)} \right)}{2 \left(n-1/24\right)^{3/2}} \right) \\ &= \frac{1}{\pi \sqrt{2}}\sum_{k=1}^{\infty}\sqrt{k}A_k(n) \bigg( \frac{\pi \exp\left( -\frac{\pi}{k} \sqrt{\frac{2}{3}( n-1/24)} \right) }{2 \sqrt{6} k \left(n-1/24\right)}\\ &\phantom{saasasdasasdadd}+\frac{\pi \exp\left( \frac{\pi}{k} \sqrt{\frac{2}{3}( n-1/24)} \right) }{2 \sqrt{6} k \left(n-1/24\right)} +\frac{\exp\left(- \frac{\pi}{k} \sqrt{\frac{2}{3}( n-1/24)} \right) }{4 \left(n-1/24\right)^{3/2}}-\frac{\exp\left( \frac{\pi}{k} \sqrt{\frac{2}{3}( n-1/24)} \right) }{4 \left(n-1/24\right)^{3/2}} \bigg) \end{align*} If \(S_m(n)\) denotes the partial sum of this series containing just the first \(m\) terms of this series, then \begin{multline*} S_2(n)=\frac{\sqrt{6} \exp\left(-\frac{1}{6} \pi \left(\sqrt{24 n-1}+6 i n\right)\right) }{\pi (24 n-1)^{3/2}} \bigg( \exp\left(\frac{1}{4} \pi \sqrt{24 n-1}\right) \left(\pi \sqrt{24 n-1}-12\right)\\ +\sqrt{2} \exp\left(\frac{1}{3} \pi \left(\sqrt{24 n-1}+3 i n\right)\right) \left(\pi \sqrt{24 n-1}-6\right)\\ +\sqrt{2} \exp\left(i \pi n\right ) \left(\pi \sqrt{24 n-1}+6\right)+ \exp\left(\frac{1}{12} \pi \sqrt{24 n-1}\right) \left(\pi \sqrt{24 n-1}+12\right) \bigg) \end{multline*} The table below shows how \(S_2(n)\) compares with \(p(n)\) for\(1\leq n \leq 60\): $$ \begin{array}{c|c|c||c|c|c} n&S_2(n)&p(n)&n&S_2(n)&p(n)\\ \hline 1 & 1.002968426 & 1 & 31 & 6842.198846 & 6842 \\ 2 & 2.080862500 & 2 & 32 & 8349.208295 & 8349 \\ 3 & 2.934087196 & 3 & 33 & 10142.47286 & 10143 \\ 4 & 5.029621840 & 5 & 34 & 12310.41065 & 12310 \\ 5 & 7.027886465 & 7 & 35 & 14883.32492 & 14883 \\ 6 & 10.93248804 & 11 & 36 & 17976.23834 & 17977 \\ 7 & 15.04202980 & 15 & 37 & 21637.14011 & 21637 \\ 8 & 22.05549058 & 22 & 38 & 26015.61334 & 26015 \\ 9 & 29.86527184 & 30 & 39 & 31184.43678 & 31185 \\ 10 & 42.06974433 & 42 & 40 & 37338.04033 & 37338 \\ 11 & 56.12189161 & 56 & 41 & 44583.47436 & 44583 \\ 12 & 76.78246714 & 77 & 42 & 53173.28799 & 53174 \\ 13 & 101.0526947 & 101 & 43 & 63261.37106 & 63261 \\ 14 & 135.1956827 & 135 & 44 & 75175.48068 & 75175 \\ 15 & 175.7833175 & 176 & 45 & 89132.89927 & 89134 \\ 16 & 231.0216744 & 231 & 46 & 105558.5458 & 105558 \\ 17 & 297.1299954 & 297 & 47 & 124754.7740 & 124754 \\ 18 & 384.8296001 & 385 & 48 & 147271.8051 & 147273 \\ 19 & 490.1450324 & 490 & 49 & 173525.3089 & 173525 \\ 20 & 627.0574026 & 627 & 50 & 204226.7970 & 204226 \\ 21 & 791.6726192 & 792 & 51 & 239941.8813 & 239943 \\ 22 & 1002.201411 & 1002 & 52 & 281589.2818 & 281589 \\ 23 & 1255.275068 & 1255 & 53 & 329931.8398 & 329931 \\ 24 & 1574.598087 & 1575 & 54 & 386153.9376 & 386155 \\ 25 & 1958.005883 & 1958 & 55 & 451276.4506 & 451276 \\ 26 & 2436.344854 & 2436 & 56 & 526823.6652 & 526823 \\ 27 & 3009.656782 & 3010 & 57 & 614152.3826 & 614154 \\ 28 & 3718.078221 & 3718 & 58 & 715220.8720 & 715220 \\ 29 & 4565.300232 & 4565 & 59 & 831821.2893 & 831820 \\ 30 & 5603.547880 & 5604 & 60 & 966464.8362 & 966467 \\ \end{array} $$ The next table shows convergence to \(p(1000)=24061467864032622473692149727991\), by showing the difference \(p(1000)-S_m(1000)\) for \(1\leq m \leq 12\): $$ \begin{array}{c|c} m&p(1000)-S_m(1000)\\ \hline 1 & 4.089739935\times 10^{13} \\ 2 & -3.025453876\times 10^7 \\ 3 & 83598.74715 \\ 4 & 2338.485525 \\ 5 & -94.92966201 \\ 6 & -15.02962078 \\ 7 & 2.693261523 \\ 8 & -0.04555238099 \\ 9 & -0.1867222589 \\ 10 & -0.3710052683 \\ 11 & 0.01064119138 \\ 12 & -0.01947875745 \\ \end{array} $$ Hardy–Ramanujan-Rademacher-type series for various restricted partition functions have also been developed.

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