OFFSET
0,2
COMMENTS
GENERAL BINOMIAL IDENTITY.
When p=2, q=3, this sequence illustrates the following identity:
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
which is a special case of the more general binomial identity:
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.
FORMULA
E.g.f.: Sum_{n>=0} (2^n + 3^n)^n * exp( (2^n + 3^n)*x ) * x^n / n!.
O.g.f.: Sum_{n>=0} (2^n + 3^n)^n * x^n / (1 - (2^n + 3^n)*x)^(n+1). - Paul D. Hanna, Jul 13 2019
FORMULAS FOR TERMS.
a(n) = Sum_{k=0..n} C(n,k)*(2^k + 3^k)^n.
a(n) = Sum_{k=0..n} C(n,k)*(1 + 2^(n-k)*3^k)^n.
a(n) = Sum_{k=0..n} C(n,k)*A007689(k)^n.
a(n) = Sum_{k=0..n} C(n,k)*A094617(n,k)^n.
a(n) ~ 3^(n^2). - Vaclav Kotesovec, Jul 14 2019
EXAMPLE
E.g.f.: A(x) = 1 + 7*x + 223*x^2/2! + 49849*x^3/3! + 94705663*x^4/4! + 1616229320497*x^5/5! + 251286598125520183*x^6/6! + 357716675257916544062689*x^7/7! + 4670472774542449929397808845183*x^8/8! + ...
such that
A(x) = exp(2*x) + (2+3)*exp((2+3)*x)*x + (2^2+3^2)^2*exp(2^2+3^2)*x)*x^2/2! + (2^3+3^3)^3*exp(2^3+3^3)*x)*x^3/3! + (2^4+3^4)^4*exp(2^4+3^4)*x)*x^4/4! + ...
ORDINARY GENERATING FUNCTION.
O.g.f.: B(x) = 1 + 7*x + 223*x^2 + 49849*x^3 + 94705663*x^4 + 1616229320497*x^5 + 251286598125520183*x^6 + 357716675257916544062689*x^7 + ...
such that
B(x) = 1/(1-2*x) + (2+3)*x/(1 - (2+3)*x)^2 + (2^2+3^2)^2*x^2/(1 - (2^2+3^2)*x)^3 + (2^3+3^3)^3*x^3/(1 - (2^3+3^3)*x)^4 + (2^4+3^4)^4*x^4/(1 - (2^4+3^4)*x)^5 + ...
ILLUSTRATION OF TERMS.
a(1) = 2 + 5 = 3 + 4 = 7 ;
a(2) = 2^2 + 2*5^2 + 13^2 = 5^2 + 2*7^2 + 10^2 = 223 ;
a(3) = 2^3 + 3*5^3 + 3*13^3 + 35^3 = 9^3 + 3*13^3 + 3*19^3 + 28^3 = 49849.
MATHEMATICA
Table[Sum[Binomial[n, k]*(2^k + 3^k)^n, {k, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Jul 14 2019 *)
PROG
(PARI) {a(n)=local(p=2, q=3, s=1, t=1, u=1, v=1);
sum(k=0, n, binomial(n, k)*(s*p^k + t*q^k)^(n-k)*(u*p^k + v*q^k)^k)}
/* right side of the general binomial identity: */
{a(n)=local(p=2, q=3, s=1, t=1, u=1, v=1);
sum(k=0, n, binomial(n, k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 09 2008, revised Mar 11 2008
STATUS
approved