A276289 Expansion of x*(1 + x)/(1 - 2*x)^3.

0, 1, 7, 30, 104, 320, 912, 2464, 6400, 16128, 39680, 95744, 227328, 532480, 1232896, 2826240, 6422528, 14483456, 32440320, 72220672, 159907840, 352321536, 772800512, 1688207360, 3674210304, 7969177600, 17230200832, 37144756224, 79859548160, 171261820928, 366414397440

Offset: 0

Author

Ilya Gutkovskiy, Aug 27 2016

Keywords

• nonn
• easy

Comments

Binomial transform of pentagonal numbers (A000326).

More generally, the binomial transform of k-gonal numbers is n*Hypergeometric2F1(k/(k-2),1-n;2/(k-2);-1), where Hypergeometric2F1(a,b;c;x) is the hypergeometric function.

Coefficients in the hypergeometric series identity 1 - 7*x/(x + 6) + 30*x*(x - 1)/((x + 6)*(x + 8)) - 104*x*(x - 1)*(x - 2)/((x + 6)*(x + 8)*(x + 10)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A077616 and A084901. - Peter Bala, May 30 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Pentagonal Number
• Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Crossrefs

Cf. A000326, A084901.

Cf. A001793 (binomial transform of triangular numbers), A001788 (binomial transform of squares), A084899 (binomial transform of heptagonal numbers).

Programs

GAP

List([0..40], n-> 2^(n-3)*n*(3*n+1)); # G. C. Greubel, Jun 02 2019

Magma

[2^(n-3)*n*(3*n+1): n in [0..40]]; // G. C. Greubel, Jun 02 2019

Maple

a:=series(x*(1+x)/(1-2*x)^3,x=0,31): seq(coeff(a,x,n),n=0..40); # Paolo P. Lava, Mar 27 2019

Mathematica

LinearRecurrence[{6, -12, 8}, {0, 1, 7}, 40]
Table[2^(n - 3) n (3 n + 1), {n, 0, 40}]

PARI

concat(0, Vec(x*(1+x)/(1-2*x)^3 + O(x^40))) \\ Altug Alkan, Aug 27 2016

Sage

[2^(n-3)*n*(3*n+1) for n in (0..40)] # G. C. Greubel, Jun 02 2019

Formula

O.g.f.: x*(1 + x)/(1 - 2*x)^3.

E.g.f.: x*(2 + 3*x)*exp(2*x)/2.

a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3).

a(n) = Sum_{k = 0..n} binomial(n,k)*k*(3*k - 1)/2.

a(n) = 2^(n-3)*n*(3*n + 1).

Sum_{n>=1} 1/a(n) = 8*(-3*2^(1/3)*Hypergeometric2F1(1/3,1/3;4/3;-1) + 3 + log(2)) = 1.1906948190529335181687...