A308399 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k*(4*k + 1)).

1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 3, 1, 3, 3, 2, 6, 3, 4, 8, 4, 9, 9, 6, 15, 10, 12, 20, 12, 22, 23, 18, 35, 26, 30, 46, 32, 51, 54, 45, 76, 62, 71, 99, 76, 111, 117, 104, 160, 136, 154, 205, 167, 230, 244, 223, 319, 286, 319, 406, 349, 456, 484, 458, 619, 570, 632, 779, 695

Offset

0,9

Comments

Number of partitions of n into parts congruent to {0, 3, 5} mod 8.

Convolution inverse of A244465.

Links

Ludovic Schwob, Table of n, a(n) for n = 0..10000

Formula

G.f.: 1 / Sum_{k>=0} (-x)^A074378(k).

G.f.: Product_{k>=1} 1 / ((1 - x^(8*k - 5)) * (1 - x^(8*k - 3)) * (1 - x^(8*k))).

G.f.: ( Sum_{k>=0} A000041(k)*(-x)^k ) / ( Sum_{k>=0} A000009(2*k)*(-x)^k ).

a(n) ~ sqrt(sqrt(2) - 1) * exp(sqrt(n)*Pi/2) / (2^(9/4)*n). - Vaclav Kotesovec, May 25 2019

a(n) = a(n-3) + a(n-5) - a(n-14) - a(n-18) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 5, 14, 18, ... is the sequence A074378. - Ludovic Schwob, Aug 04 2021

Example

For n=23 the a(23)=6 solutions are 3+3+3+3+3+3+5, 3+3+3+3+3+8, 3+3+3+3+11, 3+5+5+5+5, 5+5+5+8, and 5+5+13.

Mathematica

nmax = 68; CoefficientList[Series[1/Sum[(-x)^(k (4 k + 1)), {k, -nmax, nmax}], {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[Product[1/((1 - x^(8 k - 5)) (1 - x^(8 k - 3)) (1 - x^(8 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[Sum[PartitionsP[k] (-x)^k, {k, 0, nmax}]/Sum[PartitionsQ[2 k] (-x)^k, {k, 0, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000009, A000041, A006950, A007742, A047622, A074378, A195850, A244465, A308400.

Sequence in context: A293312 A136405 A210871 * A287601 A035667 A092865

Adjacent sequences: A308396 A308397 A308398 * A308400 A308401 A308402

Keyword

nonn

Author

Ilya Gutkovskiy, May 24 2019

Status

approved