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A264400 Number of parts of even multiplicities in all the partitions of n. 3
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 (list; graph; refs; listen; history; text; internal format)
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 * A225367 A283479 A087237

Adjacent sequences:  A264397 A264398 A264399 * A264401 A264402 A264403

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 21 2015

STATUS

approved

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Last modified June 25 09:48 EDT 2019. Contains 324347 sequences. (Running on oeis4.)