A036017 Number of partitions of n into parts not of form 4k+2, 12k, 12k+1 or 12k-1.

1, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 4, 6, 6, 7, 10, 12, 13, 15, 19, 23, 26, 29, 36, 44, 48, 54, 66, 77, 86, 98, 115, 134, 150, 169, 197, 227, 253, 285, 329, 375, 418, 470, 536, 607, 678, 757, 858, 969, 1076, 1200, 1353, 1516, 1683, 1873, 2098, 2343, 2596, 2878, 3211

Offset

0,8

Comments

Case k=3,i=1 of Gordon/Goellnitz/Andrews Theorem.

Also number of partitions in which no odd part is repeated, with no part of size less than or equal to 2 and where differences between parts at distance 2 are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.

Euler transform of period 12 sequence [0,0,1,1,1,0,1,1,1,0,0,0,...]. - Michael Somos, Jun 28 2004

References

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.

Links

Table of n, a(n) for n=0..59.

Formula

a(n) ~ exp(Pi*sqrt(n/3)) * sqrt(2*sqrt(3) - 3) / (12 * n^(3/4)). - Vaclav Kotesovec, May 10 2018

Mathematica

nmax = 60; CoefficientList[Series[Product[(1 - x^(4*k - 2))*(1 - x^(12*k))*(1 - x^(12*k+1-12))*(1 - x^(12*k- 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 10 2018 *)

Prog

(PARI) {a(n)=if(n<0, 0, polcoeff(1/prod(k=1, n, 1-([0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0][(k-1)%12+1])*x^k, 1+x*O(x^n)), n))} /* Michael Somos, Jun 28 2004 */

Crossrefs

Sequence in context: A029067 A048460 A351058 * A029066 A174522 A327719

Adjacent sequences: A036014 A036015 A036016 * A036018 A036019 A036020

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved