A081040 5th binomial transform of (1,4,0,0,0,0,...).

1, 9, 65, 425, 2625, 15625, 90625, 515625, 2890625, 16015625, 87890625, 478515625, 2587890625, 13916015625, 74462890625, 396728515625, 2105712890625, 11138916015625, 58746337890625, 308990478515625, 1621246337890625

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 67.

Index entries for linear recurrences with constant coefficients, signature (10,-25).

Formula

a(n) = 10*a(n-1) - 25*a(n-2), a(0)=1, a(1)=9.

a(n) = (4n+5)*5^(n-1).

a(n) = Sum_{k=0..n} (k+1)*4^k*binomial(n, k).

G.f.: (1-x)/(1-5*x)^2.

E.g.f.: exp(5*x)*(1 + 4*x). - Stefano Spezia, Jan 31 2025

Mathematica

CoefficientList[Series[(1 - x) / (1 - 5 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{10, -25}, {1, 9}, 30] (* Harvey P. Dale, Jan 10 2021 *)

Prog

(Magma) [(4*n+5)*5^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013
(PARI) a(n)=(4*n+5)*5^(n-1) \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Cf. A081039, A081041.

Sequence in context: A237040 A055284 A351530 * A102902 A127534 A037548

Adjacent sequences: A081037 A081038 A081039 * A081041 A081042 A081043

Keyword

nonn,easy,changed

Author

Paul Barry, Mar 03 2003

Status

approved