A036019 Number of partitions of n into parts not of form 4k+2, 12k, 12k+5 or 12k-5.

1, 1, 1, 2, 3, 3, 4, 5, 7, 9, 10, 13, 17, 20, 23, 29, 36, 42, 49, 59, 71, 83, 96, 113, 135, 156, 179, 210, 245, 281, 322, 372, 430, 492, 559, 641, 736, 835, 945, 1077, 1226, 1385, 1562, 1768, 2000, 2251, 2527, 2845, 3205, 3591, 4016, 4504, 5047, 5634, 6283

Offset

0,4

Comments

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

Also number of partitions in which no odd part is repeated, with at most 2 parts 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 [1,0,1,1,0,0,0,1,1,0,1,0,...]. - 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..54.

Formula

a(n) ~ sqrt(3 + 2*sqrt(3)) * exp(sqrt(n/3)*Pi) / (12*n^(3/4)). - Vaclav Kotesovec, May 09 2018

Mathematica

a[n_] := Coefficient[ 1/Product[ 1 - ({1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0}[[Mod[k-1, 12] + 1]])*x^k, {k, 1, n}] + x*O[x]^n, x, n]; Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Nov 22 2013, after Michael Somos *)

Prog

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

Crossrefs

Sequence in context: A237977 A115339 A305631 * A018120 A240200 A094979

Adjacent sequences: A036016 A036017 A036018 * A036020 A036021 A036022

Keyword

nonn,easy

Author

Olivier Gérard, Dec 11 1999

Status

approved