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G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * ((1+x)^k - 1)^k.
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%I #12 Jan 20 2015 22:45:05

%S 1,1,2,3,8,21,70,263,1072,4842,23351,120478,660372,3817413,23213642,

%T 147866712,983535760,6814069842,49050260795,366092901787,

%U 2827792333274,22566873540299,185782024439055,1575592459104692,13748110774214480,123281851161743801,1134880801686963605

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

%H Paul D. Hanna, <a href="/A245464/b245464.txt">Table of n, a(n) for n = 0..200</a>

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

%F G.f.: Sum_{n>=0} (1+x)^(n^2) * x^n / (1-x + x*(1+x)^n)^(n+1). - _Paul D. Hanna_, Jan 20 2015

%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 21*x^5 + 70*x^6 + 263*x^7 +...

%e where we have the identity:

%e A(x) = 1 + x*(1 + ((1+x)-1))

%e + x^2*(1 + 2*((1+x)-1) + ((1+x)^2-1)^2)

%e + x^3*(1 + 3*((1+x)-1) + 3*((1+x)^2-1)^2 + ((1+x)^3-1)^3)

%e + x^4*(1 + 4*((1+x)-1) + 6*((1+x)^2-1)^2 + 4*((1+x)^3-1)^3 + ((1+x)^4-1)^4)

%e + x^5*(1 + 5*((1+x)-1) + 10*((1+x)^2-1)^2 + 10*((1+x)^3-1)^3 + 5*((1+x)^4-1)^4 + ((1+x)^5-1)^5) +...

%e is equal to

%e A(x) = 1 + x*(0 + (1+x))

%e + x^2*(0 + 2*(1-(1+x))*(1+x) + (1+x)^4)

%e + x^3*(0 + 3*(1-(1+x))^2*(1+x) + 3*(1-(1+x)^2)*(1+x)^4 + (1+x)^9)

%e + x^4*(0 + 4*(1-(1+x))^3*(1+x) + 6*(1-(1+x)^2)^2*(1+x)^4 + 4*(1-(1+x)^3)*(1+x)^9 + (1+x)^16)

%e + x^5*(0 + 5*(1-(1+x))^4*(1+x) + 6*(1-(1+x)^2)^3*(1+x)^4 + 4*(1-(1+x)^3)^2*(1+x)^9 + 5*(1-(1+x)^4)*(1+x)^16 + (1+x)^25) +...

%e Also,

%e A(x) = 1 + (1+x)*x/(1-x + x*(1+x))^2 + (1+x)^4*x^2/(1-x + x*(1+x)^2)^3 + (1+x)^9*x^3/(1-x + x*(1+x)^3)^4 + (1+x)^16*x^4/(1-x + x*(1+x)^4)^5 + (1+x)^25*x^5/(1-x + x*(1+x)^5)^6 + (1+x)^36*x^6/(1-x + x*(1+x)^6)^7 +...

%o (PARI) {a(n) = polcoeff(sum(m=0,n, x^m*sum(k=0,m, binomial(m,k) * ((1+x)^k - 1 +x*O(x^n))^k )) ,n)}

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

%o (PARI) {a(n) = polcoeff(sum(m=0,n, x^m*sum(k=0,m, binomial(m,k) * (1 - (1+x)^k +x*O(x^n))^(m-k) * (1+x)^(k^2) )) ,n)}

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

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

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

%Y Cf. A245465.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 22 2014