A337852 - OEIS

%I #5 Sep 27 2020 14:51:11

%S 1,5,81,4913,1185921,1160290625,4608273662721,74051159531521793,

%T 4796659837465472798721,1248862969947666168212890625,

%U 1304426412609681656861792686592001,5459157240288132828933147334116110282753,91477746675481294892349178081259839233191936001

%N a(n) = (2^(n+1) + 1)^n.

%C In general, we have the o.g.f. identity:

%C Sum_{n>=0} m^n * q^(n^2) * x^n/(1 - b*q^n*x)^(n+1)  =  Sum_{n>=0} (m*q^n + b)^n * x^n ; here,  q=2, m=2, b=1.

%C In general, we have the e.g.f. identity:

%C Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n!  =  Sum_{n>=0} (m*q^n + b)^n * x^n / n! ; here,  q=2, m=2, b=1.

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

%F E.g.f.: Sum_{n>=0} 2^(n*(n+1)) * exp(2^n*x) * x^n / n!  =  Sum_{n>=0} (2^(n+1) + 1)^n * x^n / n!.

%e O.g.f.: A(x) = 1 + 5*x + 81*x^2 + 4913*x^3 + 1185921*x^4 + 1160290625*x^5 + 4608273662721*x^6 + 74051159531521793*x^7 + 4796659837465472798721*x^8 + ...

%e where

%e A(x) = 1/(1 - x) + 2^2*x/(1 - 2*x)^2 + 2^6*x^2/(1 - 2^2*x)^3 + 2^12*x^3/(1 - 2^3*x)^4 + 2^20*x^4/(1 - 2^4*x)^5 + 2^30*x^5/(1 - 2^5*x)^6 + ...

%o (PARI) {a(n,q,m,b) = (m*q^n + b)^n}

%o for(n=0,15, print1(a(n,q=2,m=2,b=1),", "))

%o (PARI) /* E.g.f. formula: */

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

%o for(n=0,15, print1(a(n,q=2,m=2,b=1),", "))

%o (PARI) /* E.g.f. formula: */

%o {a(n,q,m,b) = n! * polcoeff( sum(k=0,n, m^k * q^(k^2) * exp(b*q^k*x +x*O(x^n)) * x^k/k!), n)}

%o for(n=0,15, print1(a(n,q=2,m=2,b=1),", "))

%Y Cf. A055601, A251657, A337851, A136516.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 26 2020