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A117956 Number of partitions of n into exactly 2 types of parts: one odd and one even. 4
0, 0, 1, 1, 4, 3, 8, 6, 13, 10, 19, 13, 26, 20, 32, 23, 41, 31, 49, 34, 58, 45, 66, 47, 76, 60, 88, 60, 96, 76, 106, 76, 122, 93, 126, 94, 140, 111, 158, 106, 163, 134, 175, 127, 196, 150, 198, 149, 212, 170, 240, 164, 238, 200, 250, 180, 284, 214, 277, 216, 292, 238 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
LINKS
D. Christopher and T. Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), #15.11.5.
N. Benyahia Tani, S. Bouroubi, and O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Bulletin du Laboratoire 03 (2015), 18-27.
FORMULA
G.f.: Sum_{i>=1}(Sum{j>=1}(x^(2*i+2*j-1)/((1-x^(2*i-1))*(1-x^(2*j)))).
Convolution of x(n) and y(n), where x(n) is the number of even divisors of n and y(n) is the number of odd divisors of n. - Vladeta Jovovic, Apr 05 2006
EXAMPLE
a(7) = 8 because we have [6,1], [5,2], [4,3], [4,1,1,1], [3,2,2], [2,2,2,1],[2,2,1,1,1] and [2,1,1,1,1,1].
MAPLE
g := add(add(x^(2*i+2*j-1)/(1-x^(2*i-1))/(1-x^(2*j)), j=1..70), i=1..70):
gser:=series(g, x=0, 70): seq(coeff(gser, x^n), n=1..67);
MATHEMATICA
With[{nmax = 80}, CoefficientList[Series[Sum[Sum[x^(2*k + 2*j - 2)/((1 - x^(2*k - 1))*(1 - x^(2*j))), {j, 1, 2*nmax}], {k, 1, 2*nmax}], {x, 0, nmax}], x]] (* G. C. Greubel, Oct 06 2018 *)
PROG
(PARI) x='x+O('x^80); concat([0, 0], Vec(sum(k=1, 100, sum(j=1, 100, x^(2*k + 2*j - 2)/((1 - x^(2*k - 1))*(1 - x^(2*j))))))) \\ G. C. Greubel, Oct 06 2018
(Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); [0, 0] cat Coefficients(R!((&+[(&+[x^(2*k + 2*j - 2)/((1 - x^(2*k - 1))*(1 - x^(2*j))): j in [1..100]]): k in [1..100]]))); // G. C. Greubel, Oct 06 2018
CROSSREFS
Sequence in context: A011451 A200089 A330686 * A241638 A360689 A110662
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 05 2006
STATUS
approved

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Last modified February 21 12:33 EST 2024. Contains 370235 sequences. (Running on oeis4.)