A247609 a(n) = Sum_{k=0..4} binomial(8,k)*binomial(n,k).

1, 9, 45, 165, 495, 1231, 2639, 5055, 8885, 14605, 22761, 33969, 48915, 68355, 93115, 124091, 162249, 208625, 264325, 330525, 408471, 499479, 604935, 726295, 865085, 1022901, 1201409, 1402345, 1627515, 1878795, 2158131, 2467539, 2809105, 3184985, 3597405

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

G.f.: (1 + 4*x + 10*x^2 + 20*x^3 + 35*x^4)/(1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = (12 - 58*n + 217*n^2 - 98*n^3 + 35*n^4)/12.

a(n) = 1 + 8*Binomial(n, 1) + 28*Binomial(n, 2) + 56*Binomial(n, 3) + 70*Binomial(n, 4).

Mathematica

Table[(12 - 58 n + 217 n^2 - 98 n^3 + 35 n^4)/12, {n, 0, 50}] (* or *) CoefficientList[Series[(1 + 4 x + 10 x^2 + 20 x^3 + 35 x^4)/(1 - x)^5, {x, 0, 50}], x]
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 9, 45, 165, 495}, 40] (* Harvey P. Dale, Oct 19 2024 *)

Prog

(Magma) [(12-58*n+217*n^2-98*n^3+35*n^4)/12: n in [0..40]]; /* or */ [1+8*Binomial(n, 1)+28*Binomial(n, 2)+56*Binomial(n, 3)+70*Binomial(n, 4): n in [0..40]]; /* or */ I:=[1, 9, 45, 165, 495]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];
(Sage) m=4; [sum((binomial(2*m, k)*binomial(n, k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014

Crossrefs

Cf. A005408, A056108, A247608.

Sequence in context: A341069 A126899 A162602 * A341138 A008501 A341206

Adjacent sequences: A247606 A247607 A247608 * A247610 A247611 A247612

Keyword

nonn,easy

Author

Vincenzo Librandi, Sep 22 2014

Status

approved