A035462 - OEIS

%I #45 Jun 17 2023 22:55:29

%S 1,0,0,1,0,0,1,1,0,1,1,1,1,1,2,2,1,2,3,2,2,4,4,3,4,5,5,5,6,7,8,7,8,11,

%T 10,10,13,14,14,15,17,19,20,20,24,27,26,28,33,35,35,39,44,46,48,52,58,

%U 62,63,69,78,80,83,93,100,104,111,120,130,137,143,156,169,175,185,203

%N Number of partitions of n into parts 4k-1.

%C Also, number of partitions into parts 8k+3 or 8k+7.

%C Also number of partitions of n such that 2k-1 and 2k occur with the same multiplicity. Example: a(18)=3 because we have [8,7,2,1],[6,5,4,3] and [2,2,2,2,2,2,1,1,1,1,1,1]. It is easy to find a bijection between these partitions and those described in the definition. - _Emeric Deutsch_, Apr 05 2006

%H Vaclav Kotesovec, <a href="/A035462/b035462.txt">Table of n, a(n) for n = 0..10000</a>

%H James Mc Laughlin, Andrew V. Sills, Peter Zimmer, <a href="https://doi.org/10.37236/36">Rogers-Ramanujan-Slater Type Identities </a>, arXiv:1901.00946 [math.NT]

%F G.f.: 1/Product_{j>=1} (1 - x^(4*j-1)). - _Emeric Deutsch_, Apr 05 2006

%F G.f.: Sum_{n>=0} (x^(3*n) / Product_{k=1..n} (1 - x^(4*k))) = 1 + Sum_{n>=0} (x^(4*n+3) / Product_{k>=n} (1 - x^(4*k+3))) = 1 + Sum_{n>=0} (x^(4*n+3) / Product_{k=0..n} (1 - x^(4*k+3))). - _Joerg Arndt_, Apr 08 2011

%F a(n) ~ Pi^(3/4) * exp(Pi*sqrt(n/6)) / (Gamma(1/4) * 2^(13/8) * 3^(3/8) * n^(7/8)) * (1 + (Pi/(96*sqrt(6)) - 21*sqrt(3/2)/(16*Pi)) / sqrt(n)). - _Vaclav Kotesovec_, Feb 26 2015, extended Jan 24 2017

%F a(n) = (1/n)*Sum_{k=1..n} A050452(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 20 2017

%F From _Peter Bala_, Feb 02 2021: (Start)

%F G.f.: A(x) = Sum_{n >= 0} x^(n*(4*n-1))/Product_{k = 1..n} ( (1 - x^(4*k))*(1 - x^(4*k-1)) ). (Set z = x^3 and q = x^4 in Mc Laughlin et al., Section 1.3, Entry 7.)

%F Similarly, A(x) =  Sum_{n >= 0} x^(n*(4*n+3))/( (1 - x^3)*Product_{k = 1..n} ((1 - x^(4*k))*(1 - x^(4*k+3))) ). (End)

%e a(18)=3 because we have [15,3],[11,7] and [3,3,3,3,3,3].

%p g:=1/product(1-x^(4*i-1),i=1..50): gser:=series(g,x=0,80): seq(coeff(gser,x,n),n=1..75); # _Emeric Deutsch_, Apr 05 2006

%t nmax = 100; CoefficientList[Series[Product[1/(1-x^(4*k+3)),{k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 26 2015 *)

%t nmax = 50; kmax = nmax/4; s = Range[0, kmax]*4 - 1;

%t Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* _Robert Price_, Aug 04 2020 *)

%Y Cf. A035441-A035468, A050452.

%Y Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), this sequence (m=4), A109700 (m=5), A109702 (m=6), A109708 (m=7).

%K nonn,easy

%O 0,15

%A _Olivier Gérard_

%E Offset changed by _N. J. A. Sloane_, Apr 11 2010