A036023 Number of partitions of n into parts not of form 4k+2, 16k, 16k+7 or 16k-7.

1, 1, 1, 2, 3, 4, 5, 6, 9, 11, 13, 17, 22, 27, 32, 40, 50, 60, 71, 86, 105, 125, 146, 174, 209, 245, 285, 336, 396, 461, 534, 621, 725, 838, 963, 1113, 1287, 1477, 1689, 1938, 2224, 2538, 2888, 3293, 3755, 4265, 4830, 5478, 6215, 7024, 7923, 8947, 10098

Offset

0,4

Comments

Case k=4,i=4 of Gordon/Goellnitz/Andrews Theorem.

Also number of partitions in which no odd part is repeated, with at most 3 parts of size less than or equal to 2 and where differences between parts at distance 3 are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.

Euler transform of period 16 sequence [1,0,1,1,1,0,0,1,0,0,1,1,1,0,1,0,...]. - Michael Somos, Jul 15 2004

References

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

Links

Table of n, a(n) for n=0..52.

Formula

a(n) ~ 3^(1/4) * sqrt(2 + sqrt(2 + sqrt(2))) * exp(sqrt(3*n/2)*Pi/2) / (2^(15/4) * n^(3/4)). - Vaclav Kotesovec, May 09 2018

Mathematica

a[n_] := If[n < 0, 0, SeriesCoefficient[ 1/Product[ 1 - ({1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0}[[Mod[k - 1, 16] + 1]])*x^k, {k, 1, n}], {x, 0, n}]]; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Jul 19 2013, translated from Pari *)

Prog

(PARI) a(n)=if(n<0, 0, polcoeff(1/prod(k=1, n, 1-([1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0][(k-1)%16+1])*x^k, 1+x*O(x^n)), n))

Crossrefs

Sequence in context: A123924 A360011 A252484 * A119952 A364698 A102571

Adjacent sequences: A036020 A036021 A036022 * A036024 A036025 A036026

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved