A147599 - OEIS

%I #19 Nov 22 2020 09:00:50

%S 1,0,0,1,0,0,0,1,0,0,1,1,0,0,1,1,0,0,2,1,0,1,2,1,0,1,3,1,0,2,3,1,0,3,

%T 4,1,1,4,4,1,1,5,5,1,2,7,5,1,3,8,6,1,5,10,6,2,6,12,7,2,9,14,7,3,11,16,

%U 8,4,15,19,8,6,18,21,9,8,23,24,10,11,27,27,11,14,34,30,12,19,39,33,14,24,47

%N Expansion of Product_{i>=1} (1+x^(4*i-1)).

%C Number of partitions into distinct parts 4*k+3.

%C Convolution of A147599 and A169975 is A000700. - _Vaclav Kotesovec_, Jan 18 2017

%H Vaclav Kotesovec, <a href="/A147599/b147599.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f. sum(n>=0, x^(2*n^2+n) / prod(k=1,n, 1-x^(4*k))) - _Joerg Arndt_, Mar 10 2011.

%F a(n) ~ exp(sqrt(n/3)*Pi/2) / (4*6^(1/4)*n^(3/4)) * (1 - (3*sqrt(3)/(4*Pi) + Pi/(192*sqrt(3))) / sqrt(n)). - _Vaclav Kotesovec_, Jan 18 2017

%t nmax = 200; CoefficientList[Series[Product[(1 + x^(4*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 18 2017 *)

%t nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 4] == 3, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Jan 18 2017 *)

%Y Cf. A000700, A169975, A170956-A170965.

%Y Cf. A262928, A281243, A281244, A281245.

%K nonn

%O 0,19

%A _N. J. A. Sloane_, Aug 29 2010