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 A244755 a(n) = Sum_{k=0..n} C(n,k) * (1 + 3^k)^(n-k). 4
 1, 3, 13, 87, 985, 19563, 697573, 44195007, 4985202865, 987432857043, 344306650353853, 209169206074748967, 222262777197258910345, 409907753371580011362363, 1317924525238880964004945813, 7341603216747343890845790989967, 71176841502529490992224798115792225 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..90 FORMULA E.g.f.: Sum_{n>=0} exp((1+3^n)*x) * x^n/n!. O.g.f.: Sum_{n>=0} x^n/(1 - (1+3^n)*x)^(n+1). a(n) ~ c * 3^(n^2/4) * 2^(n+1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0,1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd. - Vaclav Kotesovec, Jan 25 2015 EXAMPLE E.g.f.: A(x) = 1 + 3*x + 13*x^2/2! + 87*x^3/3! + 985*x^4/4! + 19563*x^5/5! +... ILLUSTRATION OF INITIAL TERMS: a(1) = (1+3^0)^1 + (1+3^1)^0 = 3; a(2) = (1+3^0)^2 + 2*(1+3^1)^1 + (1+3^2)^0 = 13; a(3) = (1+3^0)^3 + 3*(1+3^1)^2 + 3*(1+3^2)^1 + (1+3^3)^0 = 87; a(4) = (1+3^0)^4 + 4*(1+3^1)^3 + 6*(1+3^2)^2 + 4*(1+3^3)^1 + (1+3^4)^0 = 985; ... MATHEMATICA Table[Sum[Binomial[n, k] * (1 + 3^k)^(n-k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 25 2015 *) PROG (PARI) {a(n) = sum(k=0, n, binomial(n, k) * (1 + 3^k)^(n-k) )} for(n=0, 25, print1(a(n), ", ")) (PARI) /* E.g.f. Sum_{n>=0} exp((1+3^n)*x)*x^n/n!" */ {a(n)=n!*polcoeff(sum(k=0, n, exp((1+3^k)*x +x*O(x^n))*x^k/k!), n)} for(n=0, 25, print1(a(n), ", ")) (PARI) /* O.g.f. Sum_{n>=0} x^n/(1 - (1+3^n)*x)^(n+1): */ {a(n)=polcoeff(sum(k=0, n, x^k/(1-(1+3^k)*x +x*O(x^n))^(k+1)), n)} for(n=0, 25, print1(a(n), ", ")) CROSSREFS Cf. A244754, A244756, A244760, A135079, A243918. Sequence in context: A174278 A001831 A196561 * A002725 A324028 A097711 Adjacent sequences:  A244752 A244753 A244754 * A244756 A244757 A244758 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 05 2014 STATUS approved

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Last modified January 17 15:12 EST 2020. Contains 330958 sequences. (Running on oeis4.)