A396091 Number of integer partitions of 2*n with reverse-alternating sum 8.

0, 0, 0, 0, 1, 1, 3, 6, 12, 21, 38, 63, 106, 169, 269, 415, 638, 956, 1425, 2087, 3035, 4353, 6204, 8741, 12241, 16983, 23422, 32050, 43618, 58953, 79279, 105978, 141001, 186606, 245892, 322469, 421198, 547790, 709765, 916039, 1178157, 1509833, 1928626, 2455438, 3116691, 3943912

Offset

0,7

Comments

In general, fixed column k > 0 of A344610 is asymptotic to 2^(k - 5/2) * 3^(k-1) * n^(k - 3/2) * exp(Pi*sqrt(2*n/3)) / ((2*k-1)! * Pi^(2*k-1)). - Vaclav Kotesovec, May 20 2026

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

G.f.: (Product_{k>0} 1/(1-q^k)) * B(q), where B(q) is the g.f. of A396075.

a(n) = Sum_{k=0..n} A396075(k) * A000041(n-k).

a(n) ~ 3*n^(5/2) * exp(Pi*sqrt(2*n/3)) / (35*2^(5/2)*Pi^7). - Vaclav Kotesovec, May 20 2026

Mathematica

nmax = 50; With[{k = 4}, colk = CoefficientList[Series[Sum[(-1)^(i-1) * x^(i*(i-1)/2 + k)/Product[(1 - x^j), {j, 1, 2*k - i}], {i, 1, 2*k} ], {x, 0, nmax}], x]]; Table[Sum[colk[[k+1]]*PartitionsP[n-k], {k, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, May 20 2026 *)

Prog

(PARI) my(N=50, q='q+O('q^N)); concat([0, 0, 0, 0], Vec(1/prod(k=1, N, 1-q^k)*sum(j=1, 8, (-1)^(j-1)*q^(j*(j-1)/2+4)/prod(k=1, 8-j, 1-q^k))))

Crossrefs

Column k=4 of A344610.

Cf. A000041, A396075.

Sequence in context: A337462 A215005 A396090 * A006330 A293636 A087503

Adjacent sequences: A396088 A396089 A396090 * A396092 A396093 A396094

Keyword

nonn

Author

Seiichi Manyama, May 17 2026

Status

approved