A077616 Binomial transform of n^2*2^n/2.

1, 10, 63, 324, 1485, 6318, 25515, 99144, 373977, 1377810, 4979799, 17714700, 62178597, 215765046, 741360195, 2525407632, 8537599665, 28669116186, 95692860783, 317684800980, 1049522104701, 3451916556990, 11307641812443

Offset: 1

Author

Karol A. Penson, Nov 12 2002

Keywords

• nonn
• easy

Comments

With a leading zero, this is second binomial transform of the hexagonal numbers A000384 (with leading zero). - Paul Barry, Jun 09 2003

Coefficients in the hypergeometric series identity 1 - 10*x/(x + 9) + 63*x*(x - 1)/((x + 9)*(x + 12)) - 324*x*(x - 1)*(x - 2)/((x + 9)*(x + 12)*(x + 15)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A276289 and A084901. - Peter Bala, May 30 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Crossrefs

Cf. A084901, A276289.

Programs

GAP

List([1..30], n-> 3^(n-2)*n*(1+2*n)) # G. C. Greubel, Jun 03 2019

Magma

[3^(n-2)*n*(1+2*n): n in [1..30]]; // G. C. Greubel, Jun 03 2019

Mathematica

LinearRecurrence[{9, -27, 27}, {1, 10, 63}, 30] (* Jean-François Alcover, May 23 2016 *)

PARI

a(n)=n*(2*n+1)*3^(n-2) \\ Charles R Greathouse IV, Mar 19 2017

Sage

[3^(n-2)*n*(1+2*n) for n in (1..30)] # G. C. Greubel, Jun 03 2019

Formula

E.g.f: x*(1+2*x)*exp(3*x).

O.g.f: ((1/3)*x^(3/4)*3^(3/4)/(-(3*x+1)/(3*x-1)+1)^(1/4))*(-(3*x+1)/(3*x-1)-1)^(1/4)*hypergeom([ -1, 2], [3/2], 3*x/(3*x-1))/(3*x-1)^2, which can also be represented as associated Legendre function: 1/6*x^(3/4)*Pi^(1/2)*3^(3/4)*LegendreP(1, -1/2, (3*x+1)/(1-3*x))/(3*x-1)^2.

G.f.: x*(1+x)/(1-3*x)^3. - Paul Barry, Jun 09 2003

a(n) = n*(2*n+1)*3^(n-2). - Paul Barry, Jul 24 2003