A393989 - OEIS

A393989

Expansion of g.f.: Product_{k>=1} (1 + x^(k*(k+5)/2)).

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 2, 0, 1, 0, 1, 2, 0, 1, 1, 2, 0, 0, 3, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 3, 0, 1, 3, 1, 1, 3, 2, 3, 2, 2, 2, 3, 1, 0, 6, 1, 1, 4, 2, 4, 1, 4, 3, 3, 3, 1, 6, 1, 2, 6, 2, 3

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OFFSET

0,26

COMMENTS

In general, if b > 0, b is odd and g.f. is Product_{k>=1} (1 + x^(k*(k+b)/2)), then a(n) ~ exp(3*Pi^(1/3) * ((sqrt(2)-1)*Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) * ((sqrt(2)-1)*Zeta(3/2))^(1/3) / (2^(b/2 + 7/6) * sqrt(3) * Pi^(1/3) * n^(5/6)).

a(n) is the number of partitions of n into distinct parts of the form k*(k+5)/2.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

FORMULA

a(n) ~ exp(3*Pi^(1/3) * ((sqrt(2)-1)*Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) * ((sqrt(2)-1)*Zeta(3/2))^(1/3) / (2^(11/3) * sqrt(3) * Pi^(1/3) * n^(5/6)).

MATHEMATICA

nmax = 200; CoefficientList[Series[Product[1 + x^(k*(k+5)/2), {k, 1, Floor[Sqrt[25+8*nmax]/2]}], {x, 0, nmax}], x]

CROSSREFS

Cf. A024940, A393987, A393988.

Sequence in context: A271223 A260944 A101670 * A378373 A351978 A219491

Adjacent sequences: A393986 A393987 A393988 * A393990 A393991 A393992

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 06 2026

STATUS

approved