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%I #13 Jul 13 2019 23:49:28
%S 1,5,47,1193,113855,46857665,83540629607,629692415941433,
%T 19653639560140008575,2505063418700072099312705,
%U 1292764583816731772891346438887,2687238342732260436646020885678131993,22431974111110989403331425804893720873764255
%N E.g.f.: A(x) = Sum_{n>=0} (1+2^n)^n * exp((1+2^n)*x) * x^n/n!.
%C GENERAL BINOMIAL IDENTITY.
%C When p=1, q=2, this sequence illustrates the following identity.
%C Given e.g.f.: Sum_{n>=0} (p^n+q^n)^n*exp((p^n+q^n)*x)*x^n/n! = Sum_{n>=0} a(n)*x^n/n!,
%C then a(n) = Sum_{k=0..n} C(n,k)*(p^k + q^k)^n = Sum_{k=0..n} C(n,k)*(1 + p^(n-k)*q^k)^n;
%C which is a special case of the more general binomial identity:
%C Sum_{k=0..n} C(n,k)*(s*p^k + t*q^k)^(n-k) * (u*p^k + v*q^k)^k = Sum_{k=0..n} C(n,k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k.
%F GENERATING FUNCTIONS.
%F E.g.f.: Sum_{n>=0} (1 + 2^n)^n * exp( (1 + 2^n)*x ) * x^n / n!.
%F O.g.f.: Sum_{n>=0} (1 + 2^n)^n * x^n / (1 - (1 + 2^n)*x)^(n+1). - _Paul D. Hanna_, Jul 13 2019
%F FORMULAS FOR TERMS.
%F a(n) = Sum_{k=0..n} binomial(n,k) * (1 + 2^k)^n.
%F a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Jun 25 2013
%e E.g.f.: A(x) = 1 + 5*x + 47*x^2/2! + 1193*x^3/3! + 113855*x^4/4! +...
%e where
%e A(x) = exp((1+1)*x) + (1+2)*exp((1+2)*x)*x + (1+2^2)^2*exp((1+2^2)*x)*x^2/2! + (1+2^3)^3*exp((1+2^3)*x)*x^3/3! +...
%e or, equivalently,
%e A(x) = exp(2*x) + 3*exp(3*x)*x + 5^2*exp(5*x)*x^2/2! + 9^3*exp(9*x)*x^3/3! + 17^4*exp(17*x)*x^4/4! + 33^5*exp(33*x)*x^5/5! +...
%e Illustrate the formula for the terms:
%e a(1) = (1+1) + (1+2) = 5 ;
%e a(2) = (1+1)^2 + 2*(1+2)^2 + (1+2^2)^2 = 2^2 + 2*3^2 + 5^2 = 47 ;
%e a(3) = (1+1)^3 + 3*(1+2)^3 + 3*(1+2^2)^3 + (1+2^3)^3 = 2^3 + 3*3^3 + 3*5^3 + 9^3 = 1193 ;
%e a(4) = (1+1)^4 + 4*(1+2)^4 + 6*(1+2^2)^4 + 4*(1+2^3)^4 + (1+2^4)^4 = 2^4 + 4*3^4 + 6*5^4 + 4*9^4 + 17^4 = 113855.
%t Table[Sum[Binomial[n,k]*(1+2^k)^n, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 25 2013 *)
%o (PARI) {a(n)=local(p=1, q=2);n!*polcoeff(sum(m=0,n,(p^m+q^m)^m*exp((p^m+q^m+x*O(x^n))*x)*x^m/m!),n)}
%o (PARI) {a(n)=local(p=1, q=2, s=1, t=1, u=1, v=1);
%o sum(k=0, n, binomial(n, k)*(s*p^k + t*q^k)^(n-k)*(u*p^k + v*q^k)^k)}
%o (PARI) /* right side of the general binomial identity: */
%o {a(n)=local(p=1, q=2, s=1, t=1, u=1, v=1);
%o sum(k=0, n, binomial(n, k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k)}
%Y Cf. A138247, A196457.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 02 2011
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