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A131942 Number of partitions of n in which each odd part has odd multiplicity. 8
1, 1, 1, 3, 3, 6, 6, 11, 13, 21, 24, 35, 44, 59, 74, 99, 126, 158, 202, 250, 320, 392, 495, 598, 758, 908, 1134, 1358, 1685, 2003, 2466, 2925, 3576, 4234, 5129, 6064, 7308, 8612, 10305, 12135, 14443, 16963, 20085, 23548, 27754, 32482, 38105, 44503, 52042 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Brian Drake and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 101 terms from Brian Drake)

Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

FORMULA

G.f.: Product_{n>=1} (1+q^(2n-1)-q^(4n-2))/((1-q^(2n))(1-q^(4n-2))).

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

EXAMPLE

a(5)=6 because 5, 4+1, 3+2, 2+2+1, 2+1+1+1 and 1+1+1+1+1 have all odd parts with odd multiplicity. The partition 3+1+1 is the partition of 5 which is not counted.

MAPLE

A:= series(product( 1/(1-q^(2*n)) *(1+q^(2*n-1)-q^(4*n-2))/(1-q^(4*n-2)), n=1..15), q, 25): seq(coeff(A, q, i), i=0..24);

MATHEMATICA

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

CROSSREFS

Cf. A000041, A015128, A006950, A046682.

Sequence in context: A088528 A220153 A219627 * A240204 A200905 A280510

Adjacent sequences:  A131939 A131940 A131941 * A131943 A131944 A131945

KEYWORD

easy,nonn

AUTHOR

Brian Drake, Jul 30 2007

STATUS

approved

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Last modified February 18 04:48 EST 2020. Contains 332011 sequences. (Running on oeis4.)