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G.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n * x^n.
2

%I #7 Sep 20 2020 01:14:24

%S 1,1,2,4,9,23,62,179,549,1773,6003,21233,78187,298894,1183387,4842221,

%T 20438964,88849325,397183838,1823456223,8587051052,41434641992,

%U 204654311282,1033757421996,5335693879201,28118977852767,151192761513229,828884087889407

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

%F G.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n * x^n.

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

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 23*x^5 + 62*x^6 + 179*x^7 + 549*x^8 + 1773*x^9 + 6003*x^10 + 21233*x^11 + 78187*x^12 + ...

%e where

%e A(x) = 1 + (1 + x*(1+x))*x + (1 + x*(1+x)^2)^2*x^2 + (1 + x*(1+x)^3)^3*x^3 + (1 + x*(1+x)^4)^4*x^4 + ... + (1 + x*(1+x)^n)^n*x^n + ...

%e also

%e A(x) = 1/(1 - x) + (1+x)*x^2/(1 - x*(1+x))^2 + (1+x)^4*x^4/(1 - x*(1+x)^2)^3 + (1+x)^9*x^6/(1 - x*(1+x)^3)^4 + (1+x)^16*x^8/(1 - x*(1+x)^4)^5 + (1+x)^25*x^10/(1 - x*(1+x)^5)^6 + ... + (1+x)^(n^2)*x^(2*n)/(1 - x*(1+x)^n)^(n+1) + ...

%t nmax = 30; CoefficientList[Series[Sum[(1 + x*(1+x)^k)^k * x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 17 2020 *)

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

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

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

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

%Y Cf. A337720.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 17 2020