A100847 -id:A100847 - cp's OEIS frontend

A263401 Expansion of Product_{k>=1} (1 + x^k - x^(2*k)).

1, 1, 0, 2, 0, 1, 3, 1, 1, 2, 6, 1, 4, 2, 5, 10, 5, 4, 9, 7, 8, 21, 9, 13, 13, 19, 13, 27, 32, 23, 29, 33, 27, 45, 37, 45, 79, 49, 57, 68, 82, 67, 101, 83, 109, 155, 124, 113, 174, 148, 171, 196, 215, 198, 262, 310, 269, 330, 314, 342, 414, 430, 393, 536, 493

Offset: 0

Vaclav Kotesovec, Jan 03 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000726, A002390, A055922, A100847, A109389, A162891, A266686.

nmax = 80; CoefficientList[Series[Product[1+x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[3]] = -1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] - If[j < 2*k, 0, p[[j - 2*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

a(n) ~ sqrt(log(phi)) * phi^sqrt(8*n) / (2^(3/4)*sqrt(Pi)*n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

A102247 Number of partitions of n in which each odd part has odd multiplicity and each even part has even multiplicity.

Original entry on oeis.org

1, 1, 0, 2, 2, 3, 2, 4, 7, 8, 8, 10, 17, 17, 20, 26, 39, 39, 46, 56, 77, 85, 96, 116, 154, 172, 190, 234, 289, 328, 364, 440, 532, 610, 670, 808, 957, 1091, 1204, 1432, 1675, 1905, 2110, 2476, 2867, 3255, 3608, 4184, 4837, 5451, 6050, 6960, 7980, 8961, 9972, 11370

Offset: 0

Vladeta Jovovic, Aug 16 2007

			a(7) = 4 because we have 7, 322, 22111 and 1111111.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A055922, A117958, A130126, A131942, A100847.

g:=product((1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)),i=1..40): gser:=series(g,x=0, 60): seq(coeff(gser,x,n),n=0..55); # Emeric Deutsch, Aug 23 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i+j, 2)=0, b(n-i*j, i-1), 0), j=1..n/i)
       +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1) - x^(4*k-2))/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

G.f.: Product_{i>0} (1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)).

a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((Pi^2/3 + 4*log(phi)^2)*n)) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

More terms from Emeric Deutsch, Aug 23 2007

cp's OEIS Frontend

A263401 Expansion of Product_{k>=1} (1 + x^k - x^(2*k)).

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