A393996 Number of integer partitions of 2*n with reverse-alternating sum 4.

0, 0, 1, 1, 3, 6, 12, 20, 35, 56, 91, 140, 215, 319, 473, 682, 980, 1383, 1940, 2683, 3692, 5021, 6794, 9108, 12148, 16078, 21182, 27718, 36115, 46788, 60368, 77498, 99119, 126202, 160129, 202365, 254911, 319943, 400360, 499337, 621038, 770080, 952380, 1174575, 1445056

Offset

0,5

Links

Vaclav Kotesovec, 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 A343941.

a(n) = Sum_{k=0..n} A343941(k) * A000041(n-k).

a(n) ~ n^(1/2) * exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi^3). - Vaclav Kotesovec, May 20 2026

Mathematica

nmax = 50; With[{k = 2}, 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], Vec(1/prod(k=1, N, 1-q^k)*sum(j=1, 4, (-1)^(j-1)*q^(j*(j-1)/2+2)/prod(k=1, 4-j, 1-q^k))))

Crossrefs

Column k=2 of A344610.

Cf. A000041, A343941.

Sequence in context: A320612 A320613 A006128 * A247661 A079983 A028926

Adjacent sequences: A393993 A393994 A393995 * A393997 A393998 A393999

Keyword

nonn

Author

Seiichi Manyama, May 17 2026

Status

approved