A054440 Number of ordered pairs of partitions of n with no common parts.
1, 0, 2, 4, 12, 16, 48, 60, 148, 220, 438, 618, 1302, 1740, 3216, 4788, 8170, 11512, 19862, 27570, 45448, 64600, 100808, 141724, 223080, 307512, 465736, 652518, 968180, 1334030, 1972164, 2691132, 3902432, 5347176, 7611484, 10358426, 14697028, 19790508, 27691500
Offset: 0
Examples
a(3)=4 because of the 4 pairs of partitions of 3: (3,21),(3,111),(21,3),(111,3).
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..5000
- Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, Doron Zeilberger, A pentagonal number sieve, J. Combin. Theory Ser. A 82 (1998), no. 2, 186-192.
- Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
- Wikipedia, Pentagonal number theorem
Crossrefs
Programs
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Haskell
a054440 = sum . zipWith (*) a087960_list . map a001255 . a260672_row -- Reinhard Zumkeller, Nov 15 2015
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Maple
with(combinat): p1 := sum(numbpart(n)^2*x^n, n=0..500): it := p1*product((1-x^i), i=1..500): s := series(it, x, 500): for i from 0 to 100 do printf(`%d,`,coeff(s,x,i)) od:
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Mathematica
nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]^2*x^k, {k, 0, nmax}]/Sum[PartitionsP[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
Formula
G.f.: Sum[p(n)^2*x^n]/Sum[p(n)*x^n], with p(n)=number of partitions of n.
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n)) / (64 * 2^(1/4) * n^(7/4)). - Vaclav Kotesovec, May 20 2018
a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k / (1 - x^k) + y^k / (1 - y^k)). - Ilya Gutkovskiy, Apr 24 2025
Extensions
Corrected and extended by James Sellers, May 23 2000