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

0,3

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

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

Example

a(3)=4 because of the 4 pairs of partitions of 3: (3,21),(3,111),(21,3),(111,3).

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:

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 *)

Prog

(Haskell)
a054440 = sum . zipWith (*) a087960_list . map a001255 . a260672_row
-- Reinhard Zumkeller, Nov 15 2015

Crossrefs

Cf. A000041, A001255, A001318, A087960, A260672, A260664, A260669, A304873, A304877.

Main diagonal of A284592.

Sequence in context: A132314 A053636 A181135 * A308076 A303819 A074646

Adjacent sequences: A054437 A054438 A054439 * A054441 A054442 A054443

Keyword

easy,nonn

Author

Herbert S. Wilf, May 13 2000

Extensions

Corrected and extended by James Sellers, May 23 2000

Status

approved