This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A138247 #13 Jan 31 2025 01:29:41
%S A138247 1,7,223,49849,94705663,1616229320497,251286598125520183,
%T A138247 357716675257916544062689,4670472774542449929397808845183,
%U A138247 559006854195449142958954163012808059617,612171730457531439763516750114999086563829844663,6118056385739077528636842573416061383741677666682643900049
%N A138247 E.g.f.: Sum_{n>=0} exp( (2^n+3^n)*x ) * (2^n+3^n)^n * x^n/n!.
%C A138247 GENERAL BINOMIAL IDENTITY.
%C A138247 When p=2, q=3, this sequence illustrates the following identity:
%C A138247 Sum_{k=0..n} C(n,k)*(p^k + q^k)^n =
%C A138247 Sum_{k=0..n} C(n,k)*(1 + p^(n-k)*q^k)^n
%C A138247 which is a special case of the more general binomial identity:
%C A138247 Sum_{k=0..n} C(n,k)*(s*p^k + t*q^k)^(n-k) * (u*p^k + v*q^k)^k =
%C A138247 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 A138247 E.g.f.: Sum_{n>=0} (2^n + 3^n)^n * exp( (2^n + 3^n)*x ) * x^n / n!.
%F A138247 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
%F A138247 FORMULAS FOR TERMS.
%F A138247 a(n) = Sum_{k=0..n} C(n,k)*(2^k + 3^k)^n.
%F A138247 a(n) = Sum_{k=0..n} C(n,k)*(1 + 2^(n-k)*3^k)^n.
%F A138247 a(n) = Sum_{k=0..n} C(n,k)*A007689(k)^n.
%F A138247 a(n) = Sum_{k=0..n} C(n,k)*A094617(n,k)^n.
%F A138247 a(n) ~ 3^(n^2). - _Vaclav Kotesovec_, Jul 14 2019
%e A138247 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! + ...
%e A138247 such that
%e A138247 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! + ...
%e A138247 ORDINARY GENERATING FUNCTION.
%e A138247 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 + ...
%e A138247 such that
%e A138247 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 + ...
%e A138247 ILLUSTRATION OF TERMS.
%e A138247 a(1) = 2 + 5 = 3 + 4 = 7 ;
%e A138247 a(2) = 2^2 + 2*5^2 + 13^2 = 5^2 + 2*7^2 + 10^2 = 223 ;
%e A138247 a(3) = 2^3 + 3*5^3 + 3*13^3 + 35^3 = 9^3 + 3*13^3 + 3*19^3 + 28^3 = 49849.
%t A138247 Table[Sum[Binomial[n, k]*(2^k + 3^k)^n, {k, 0, n}], {n, 0, 12}] (* _Vaclav Kotesovec_, Jul 14 2019 *)
%o A138247 (PARI) {a(n)=local(p=2,q=3,s=1,t=1,u=1,v=1);
%o A138247 sum(k=0,n,binomial(n,k)*(s*p^k + t*q^k)^(n-k)*(u*p^k + v*q^k)^k)}
%o A138247 /* right side of the general binomial identity: */
%o A138247 {a(n)=local(p=2,q=3,s=1,t=1,u=1,v=1);
%o A138247 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 A138247 Cf. A007689, A094617, A196457, A196458, A196459, A196460.
%K A138247 nonn
%O A138247 0,2
%A A138247 _Paul D. Hanna_, Mar 09 2008, revised Mar 11 2008