A035622 -id:A035622 - cp's OEIS frontend

A035621 Number of partitions of n into parts 4k and 4k+1 with at least one part of each type.

0, 0, 0, 0, 1, 1, 1, 1, 4, 4, 4, 4, 10, 11, 11, 11, 22, 25, 26, 26, 44, 51, 54, 55, 84, 98, 105, 108, 153, 178, 193, 200, 269, 313, 341, 356, 459, 531, 582, 611, 764, 880, 967, 1021, 1244, 1424, 1568, 1662, 1988, 2264, 2494, 2653, 3122, 3536, 3896, 4155

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Robert Price)

Cf. A035441-A035468, A035618-A035620, A035622-A035699.

nmax = 56; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 1;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 56; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

G.f.: (-1 + 1/Product_{k>=1} (1 - x^(4 k)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 1))). - Robert Price, Aug 16 2020

A035623 Number of partitions of n into parts 4k and 4k+3 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 1, 3, 6, 1, 3, 7, 12, 3, 7, 15, 21, 7, 16, 28, 36, 16, 31, 50, 60, 32, 57, 85, 98, 60, 100, 141, 157, 107, 169, 226, 248, 184, 276, 358, 385, 305, 442, 553, 591, 495, 691, 845, 896, 782, 1063, 1270, 1343, 1216, 1608, 1890, 1993

Offset: 1

Olivier Gérard

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A035441-A035468, A035618-A035622, A035624-A035699.

N:= 100:
P:= (-1 + 1/mul(1-x^(4*k+3), k=0..(N-3)/4))*(-1 + 1/mul(1-x^(4*k), k=1..N/4)):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Feb 23 2016

nmax = 63; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 63; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 06 2020 *)

G.f.: (-1 + 1/Product_{k>=0} (1-x^(4k+3)))*(-1 + 1/Product_{k>=1} (1-x^(4k))). - Robert Israel, Feb 23 2016

a(n) ~ exp(Pi*sqrt(n/3)) * Pi^(3/4) / (2^(5/4) * 3^(5/8) * Gamma(1/4) * n^(9/8)). - Vaclav Kotesovec, May 26 2018

cp's OEIS Frontend

A035621 Number of partitions of n into parts 4k and 4k+1 with at least one part of each type.

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