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A056970 Number of partitions of n into distinct parts congruent to 2, 4 or 5 mod 6. 8
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 (list; graph; refs; listen; history; text; internal format)
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, 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: A194328 A194304 A090701 * A212218 A321162 A008668

Adjacent sequences:  A056967 A056968 A056969 * A056971 A056972 A056973

KEYWORD

nonn,nice,easy

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified July 17 14:44 EDT 2019. Contains 325106 sequences. (Running on oeis4.)