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 A337851 #10 Sep 27 2020 14:51:40 %S A337851 1,4,36,1000,104976,45435424,82653950016,627485170000000, %T A337851 19631688197463081216,2504194578379511247798784, %U A337851 1292628144912333835229805413376,2687153475176994340820312500000000000,22431765115399782718874449007331506546282496 %N A337851 a(n) = (2^n + 2)^n. %C A337851 In general, we have the o.g.f. identity: %C A337851 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=1, b=2. %C A337851 In general, we have the e.g.f. identity: %C A337851 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=1, b=2. %F A337851 O.g.f.: Sum_{n>=0} 2^(n^2) * x^n/(1 - 2^(n+1)*x)^(n+1) = Sum_{n>=0} (2^n + 2)^n * x^n. %F A337851 E.g.f.: Sum_{n>=0} 2^(n^2) * exp(2^(n+1)*x) * x^n / n! = Sum_{n>=0} (2^n + 2)^n * x^n / n!. %F A337851 a(n) = 2^n * A165327(n) for n >= 0. %e A337851 O.g.f.: A(x) = 1 + 4*x + 36*x^2 + 1000*x^3 + 104976*x^4 + 45435424*x^5 + 82653950016*x^6 + 627485170000000*x^7 + 19631688197463081216*x^8 + ... %e A337851 where %e A337851 A(x) = 1/(1 - 2*x) + 2*x/(1 - 2^2*x)^2 + 2^4*x^2/(1 - 2^3*x)^3 + 2^9*x^3/(1 - 2^4*x)^4 + 2^16*x^4/(1 - 2^5*x)^5 + 2^25*x^5/(1 - 2^6*x)^6 + ... %o A337851 (PARI) {a(n,q,m,b) = (m*q^n + b)^n} %o A337851 for(n=0,15, print1(a(n,q=2,m=1,b=2),", ")) %o A337851 (PARI) /* E.g.f. formula: */ %o A337851 {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 A337851 for(n=0,15, print1(a(n,q=2,m=1,b=2),", ")) %o A337851 (PARI) /* E.g.f. formula: */ %o A337851 {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 A337851 for(n=0,15, print1(a(n,q=2,m=1,b=2),", ")) %Y A337851 Cf. A165327, A055601, A251657, A337852, A136516. %K A337851 nonn %O A337851 0,2 %A A337851 _Paul D. Hanna_, Sep 26 2020