A320950 - OEIS

%I #5 Oct 26 2018 22:02:58

%S 1,2,3,5,20,81,272,1144,6147,30859,158137,955988,5995439,37307475,

%T 252176301,1813873656,13149151909,99412177075,793516947530,

%U 6470733413532,54217400538306,473499984230701,4245890615280401,38948094201082823,368815668052736968,3585473523132486254,35608100771085923165,362850695679003347638,3788143752503214124895

%N G.f.: [ Sum_{n>=0} x^n * (1+x)^(n^2) ] * [ Sum_{n>=0} x^n / (1+x)^(n^2) ].

%e G.f.: A(x) = 1 + 2*x + 3*x^2 + 5*x^3 + 20*x^4 + 81*x^5 + 272*x^6 + 1144*x^7 + 6147*x^8 + 30859*x^9 + 158137*x^10 + 955988*x^11 + 5995439*x^12 + ...

%e such that A(x) = P(x) * Q(x) where

%e P(x) = 1 + x*(1+x) + x^2*(1+x)^4 + x^3*(1+x)^9 + x^4*(1+x)^16 + x^5*(1+x)^25 + x^6*(1+x)^36 + x^7*(1+x)^49 + ... + x^n * (1+x)^(n^2) + ...

%e Q(x) = 1 + x/(1+x) + x^2/(1+x)^4 + x^3/(1+x)^9 + x^4/(1+x)^16 + x^5/(1+x)^25 + x^6/(1+x)^36 + x^7/(1+x)^49 + ... + x^n / (1+x)^(n^2) + ...

%e Explicitly,

%e P(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 57*x^5 + 231*x^6 + 1023*x^7 + 4926*x^8 + 25483*x^9 + 140601*x^10 + 822422*x^11 + ... + A121689(n)*x^n + ...

%e Q(x) = 1 + x - 2*x^3 + x^4 + 11*x^5 - 19*x^6 - 86*x^7 + 365*x^8 + 581*x^9 - 7336*x^10 + 6061*x^11 + 142946*x^12 - 556061*x^13 + ...

%o (PARI) {a(n) = my(A = sum(m=0, n, x^m*(1+x + x*O(x^n))^(m^2) ) * sum(m=0, n, x^m/(1+x + x*O(x^n))^(m^2) )); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A121689, A320830.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 26 2018