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 A243918 a(n) = Sum_{k=0..n} binomial(n,k) * (1 + 2^k)^k. 6
 1, 4, 32, 814, 86600, 39560554, 75654970772, 594996059517934, 19035905851947436400, 2460857798358946973785234, 1280109151917797032199865564812, 2672783800502564772495577135824089014, 22366199286781599568269093307412768076442280 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA a(n) = Sum_{k=0..n} binomial(n,k) * (1 + 2^k)^(n-k) * 2^(k^2). O.g.f.: Sum_{n>=0} (1 + 2^n)^n * x^n / (1-x)^(n+1). O.g.f.: Sum_{n>=0} 2^(n^2) * x^n / (1 - (1+2^n)*x)^(n+1). E.g.f.: exp(x) * Sum_{n>=0} (1 + 2^n)^n * x^n / n!. a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jun 18 2014 EXAMPLE O.g.f.: A(x) = 1 + 4*x + 32*x^2 + 814*x^3 + 86600*x^4 + 39560554*x^5 +... where the g.f. may be expressed by the series identity: A(x) = 1/(1-x) + 3*x/(1-x)^2 + 5^2*x^2/(1-x)^3 + 9^3*x^3/(1-x)^4 + 17^4*x^4/(1-x)^5 + 33^5*x^5/(1-x)^6 + 65^6*x^6/(1-x)^7 +... A(x) = 1/(1-2*x) + 2*x/(1-3*x)^2 + 2^4*x^2/(1-5*x)^3 + 2^9*x^3/(1-9*x)^4 + 2^16*x^4/(1-17*x)^5 + 2^25*x^5/(1-33*x)^6 + 2^36*x^6/(1-65*x)^7 +... Illustration of initial terms: a(0) = 1; a(1) = 1 + (1+2); a(2) = 1 + 2*(1+2) + (1+2^2)^2; a(3) = 1 + 3*(1+2) + 3*(1+2^2)^2 + (1+2^3)^3; a(4) = 1 + 4*(1+2) + 6*(1+2^2)^2 + 4*(1+2^3)^3 + (1+2^4)^4; a(5) = 1 + 5*(1+2) + 10*(1+2^2)^2 + 10*(1+2^3)^3 + 5*(1+2^4)^4 + (1+2^5)^5; ... Also, by a binomial identity we have a(0) = 1; a(1) = 2 + 2; a(2) = 2^2 + 2*(1+2)*2 + 2^4; a(3) = 2^3 + 3*(1+2)^2*2 + 3*(1+2^2)*2^4 + 2^9; a(4) = 2^4 + 4*(1+2)^3*2 + 6*(1+2^2)^2*2^4 + 4*(1+2^3)*2^9 + 2^16; a(5) = 2^5 + 5*(1+2)^4*2 + 10*(1+2^2)^3*2^4 + 10*(1+2^3)^2*2^9 + 5*(1+2^4)*2^16 + 2^25; ... MATHEMATICA Table[Sum[Binomial[n, k]*(1+2^k)^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 18 2014 *) PROG (PARI) {a(n)=sum(k=0, n, binomial(n, k)*(1+2^k)^k)} for(n=0, 20, print1(a(n), ", ")) (PARI) {a(n)=sum(k=0, n, binomial(n, k)*(1+2^k)^(n-k)*2^(k^2))} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A244004, A136516. Sequence in context: A134087 A132854 A136471 * A298896 A231991 A028369 Adjacent sequences:  A243915 A243916 A243917 * A243919 A243920 A243921 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 17 2014 STATUS approved

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Last modified August 14 19:47 EDT 2020. Contains 336483 sequences. (Running on oeis4.)