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
Examples
a(18)=4 because we have [16,2], [14,4], [11,5,2] and [10,8].
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.
Programs
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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
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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
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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 *) -
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 */
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
Comments