A081901 Second binomial transform of binomial(n+5, 5).

1, 8, 49, 262, 1286, 5944, 26262, 111996, 464103, 1877904, 7446735, 29021490, 111405780, 422003520, 1579757580, 5851519704, 21468622077, 78087814776, 281798184573, 1009617794334, 3593281988754, 12710491403112, 44705999907666, 156414048864948, 544562500963779, 1887215083472448

Offset: 0

Author

Paul Barry, Mar 31 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A055852 (without leading 0).

3rd binomial transform of (1,5,10,10,5,1,0,0,0,...).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (18,-135,540,-1215,1458,-729).

Crossrefs

Cf. A000389, A055852, A081902.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x)^5/(1-3*x)^6)); // G. C. Greubel, Oct 18 2018

Mathematica

LinearRecurrence[{18, -135, 540, -1215, 1458, -729}, {1, 8, 49, 262, 1286, 5944}, 50] (* G. C. Greubel, Oct 18 2018 *)
CoefficientList[Series[(1-2x)^5/(1-3x)^6,{x,0,30}],x] (* Harvey P. Dale, Oct 22 2024 *)

PARI

x='x+O('x^30); Vec((1-2*x)^5/(1-3*x)^6) \\ G. C. Greubel, Oct 18 2018

Formula

a(n) = 3^n*(n^5 + 65*n^4 + 1385*n^3 + 11575*n^2 + 35574*n + 29160)/29160.

G.f.: (1 - 2*x)^5/(1 - 3*x)^6.

E.g.f.: (120 + 600*x + 600*x^2 + 200*x^3 + 25*x^4 + x^5)*exp(3*x)/120. - G. C. Greubel, Oct 18 2018