A290942 - OEIS

A290942

Number of partitions of n into distinct generalized pentagonal numbers (A001318).

1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 0, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 4, 3, 3, 3, 2, 5, 4, 5, 4, 2, 3, 3, 6, 6, 5, 5, 4, 5, 7, 8, 8, 7, 6, 6, 6, 8, 9, 9, 9, 7, 8, 9, 9, 11, 10, 11, 11, 10, 12, 10, 14, 15, 14, 14, 11, 13, 13, 17, 17, 14, 15, 14, 17, 20, 19, 20, 20, 20, 21, 20, 21, 21, 25, 26, 23, 22, 21, 24, 27

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OFFSET

0,8

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Eric Weisstein's World of Mathematics, Pentagonal Number

Index to sequences related to polygonal numbers

Index entries for related partition-counting sequences

FORMULA

G.f.: Product_{k>=1} (1 + x^(k*(3*k-1)/2))*(1 + x^(k*(3*k+1)/2)).

a(n) ~ ((sqrt(2) - 1)*zeta(3/2))^(1/3) * exp(Pi^(1/3) * ((3/2)*(sqrt(2) - 1)*zeta(3/2))^(2/3) * n^(1/3)) / (2^(4/3) * 3^(2/3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Mar 11 2026

EXAMPLE

a(15) = 3 because we have [15], [12, 2, 1] and [7, 5, 2, 1].

MATHEMATICA

nmax = 90; CoefficientList[Series[Product[(1 + x^(k (3 k - 1)/2)) (1 + x^(k (3 k + 1)/2)), {k, 1, Floor[Sqrt[1 + 24*nmax]/6 + 1]}], {x, 0, nmax}], x] (* tuned for efficiency by Vaclav Kotesovec, Mar 11 2026 *)

CROSSREFS

Cf. A000009, A001318, A095699, A218379, A218380, A279221, A280952, A281083.

Sequence in context: A241079 A061198 A195011 * A039801 A105821 A342936

Adjacent sequences: A290939 A290940 A290941 * A290943 A290944 A290945

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Aug 14 2017

STATUS

approved