A264400 Number of parts of even multiplicities in all the partitions of n.
0, 0, 1, 0, 3, 2, 6, 6, 15, 15, 29, 34, 58, 70, 109, 132, 199, 246, 348, 435, 601, 746, 1005, 1252, 1653, 2053, 2666, 3298, 4231, 5219, 6608, 8124, 10198, 12476, 15525, 18927, 23374, 28387, 34823, 42122, 51376, 61922, 75098, 90200, 108874, 130298, 156564, 186777, 223490, 265779, 316799
Offset: 0
Keywords
Examples
a(6) = 6 because we have [6], [5,1], [4,2], [4,1*,1], [3*,3], [3,2,1], [3,1,1,1], [2,2,2], [2*,2,1*,1], [2,1*,1,1,1], and [1*,1,1,1,1,1] (the 6 parts with even multiplicities are marked).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Programs
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Maple
g := (sum(x^(2*j)/(1+x^j), j = 1 .. 100))/(product(1-x^j, j = 1 .. 100)): gser := series(g, x = 0, 70): seq(coeff(gser, x, n), n = 0 .. 60);
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Mathematica
Needs["Combinatorica`"]; Table[Count[Last /@ Flatten[Tally /@ Combinatorica`Partitions@ n, 1], k_ /; EvenQ@ k], {n, 0, 50}] (* Michael De Vlieger, Nov 21 2015 *) Table[Sum[(1 - 2*DivisorSigma[0, 2*k] + 3*DivisorSigma[0, k]) * PartitionsP[n-k], {k, 1, n}], {n, 0, 50}] (* Vaclav Kotesovec, Jun 14 2025 *) -
PARI
{ my(n=50); Vec(sum(k=1, n, x^(2*k)/(1+x^k) + O(x*x^n)) / prod(k=1, n, 1-x^k + O(x*x^n)), -(n+1)) } \\ Andrew Howroyd, Dec 22 2017
Formula
G.f.: g(x) = (Sum_{j>=1} (x^(2j)/(1+x^j))) / Product_{k>=1} (1-x^k).
Comments