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 A337852 a(n) = (2^(n+1) + 1)^n. 1
 1, 5, 81, 4913, 1185921, 1160290625, 4608273662721, 74051159531521793, 4796659837465472798721, 1248862969947666168212890625, 1304426412609681656861792686592001, 5459157240288132828933147334116110282753, 91477746675481294892349178081259839233191936001 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS In general, we have the o.g.f. identity: 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. In general, we have the e.g.f. identity: 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. LINKS FORMULA 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. 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!. EXAMPLE 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 + ... where 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 + ... PROG (PARI) {a(n, q, m, b) = (m*q^n + b)^n} for(n=0, 15, print1(a(n, q=2, m=2, b=1), ", ")) (PARI) /* E.g.f. formula: */ {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)} for(n=0, 15, print1(a(n, q=2, m=2, b=1), ", ")) (PARI) /* E.g.f. formula: */ {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)} for(n=0, 15, print1(a(n, q=2, m=2, b=1), ", ")) CROSSREFS Cf. A055601, A251657, A337851, A136516. Sequence in context: A009634 A165435 A197443 * A280220 A280675 A209102 Adjacent sequences: A337849 A337850 A337851 * A337853 A337854 A337855 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 26 2020 STATUS approved

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Last modified December 4 23:46 EST 2022. Contains 358572 sequences. (Running on oeis4.)