A056970 Number of partitions of n into distinct parts congruent to 2, 4 or 5 mod 6.

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 11, 13, 13, 15, 16, 17, 20, 21, 23, 25, 27, 30, 33, 36, 38, 42, 45, 49, 54, 57, 62, 67, 72, 79, 85, 92, 98, 106, 114, 123, 133, 141, 152, 163, 175, 189, 202, 216, 231, 248, 265, 284, 304, 323

Offset

0,11

Comments

Also number of partitions of n into parts equal to 2,5, or 11 mod 12 (Gollnitz's theorem). Example: a(18)=4 because we have [14,2,2], [11,5,2], [5,5,2,2,2,2] and [2,2,2,2,2,2,2,2,2]. - Emeric Deutsch, Apr 18 2006

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

K. Alladi, Going beyond the partition theorem of Goellnitz

G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 101.

G. E. Andrews, K. Alladi, and B. Gordon, Generalizations and refinements of a partition theorem of Göllnitz, Journal für die reine und angewandte Mathematik (1995), Volume: 460, page 165-188.

H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. Vol. 225 (1967), 154-190.

Eric Weisstein's World of Mathematics, Göllnitz's Theorem.

Formula

From Emeric Deutsch, Apr 18 2006: (Start)

G.f.: Product_{j >= 0} (1+x^(2+6j))(1+x^(4+6j))(1+x^(5+6j)).

G.f.: 1/Product_{j >= 0} (1-x^(2+12j))(1-x^(5+12j))(1-x^(11+12j)).

(End)

Euler transform of period 12 sequence [ 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, ...]. - Michael Somos, Jul 24 2007

a(n) ~ exp(Pi*sqrt(n/6)) / (2^(25/12) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 30 2015

Example

a(18)=4 because we have [16,2], [14,4], [11,5,2] and [10,8].

Maple

g:=product((1+x^(2+6*j))*(1+x^(4+6*j))*(1+x^(5+6*j)), j=0..30): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=0..67); # Emeric Deutsch, Apr 18 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(irem(d, 12) in [2, 5, 11], d, 0)
      , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Oct 27 2015

Mathematica

max = 70; g[x_] := Product[(1+x^(2+6j))(1+x^(4+6j))(1+x^(5+6j)), {j, 0, Floor[max/6]}]; CoefficientList[ Series[g[x], {x, 0, max}], x](* Jean-François Alcover, Nov 16 2011, after Emeric Deutsch *)
a[n_] := a[n] = If[n==0, 1, Sum[Sum[If[MatchQ[Mod[d, 12], 2|5|11], d, 0], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Dec 23 2015, after Alois P. Heinz *)

Prog

(PARI) {a(n)= if(n<0, 0, polcoeff( 1/prod(k=1, n, 1-(k%3==2)*(k%12!=8)*x^k, 1+x*O(x^n)), n))} /* Michael Somos, Jul 24 2007 */
(Haskell)
a056970 n = p a047261_list n where
   p _  0     = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Nov 16 2012

Crossrefs

Cf. A047261, A096981, A097451, A098884.

Sequence in context: A090701 A339332 A330996 * A212218 A321162 A008668

Adjacent sequences: A056967 A056968 A056969 * A056971 A056972 A056973

Keyword

nonn,nice,easy

Author

Eric W. Weisstein

Status

approved