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

0,5

Comments

a(n) = Sum_{k>=0} k*A264399(n,k).

Links

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

Formula

G.f.: g(x) = (Sum_{j>=1} (x^(2j)/(1+x^j))) / Product_{k>=1} (1-x^k).

Example

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).

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);

Mathematica

Needs["Combinatorica`"]; Table[Count[Last /@ Flatten[Tally /@ Combinatorica`Partitions@ n, 1], k_ /; EvenQ@ k], {n, 0, 50}] (* Michael De Vlieger, Nov 21 2015 *)

Prog

(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

Crossrefs

Cf. A264399.

Sequence in context: A114208 A014686 A053090 * A365660 A309512 A225367

Adjacent sequences: A264397 A264398 A264399 * A264401 A264402 A264403

Keyword

nonn

Author

Emeric Deutsch, Nov 21 2015

Status

approved