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

Author

Vincenzo Librandi, Sep 22 2014

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A005408, A056108, A247608.

Programs

Magma

[(12-58*n+217*n^2-98*n^3+35*n^4)/12: n in [0..40]];

Magma

[1+8*Binomial(n, 1)+28*Binomial(n, 2)+56*Binomial(n, 3)+70*Binomial(n,4): n in [0..40]];

Magma

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]];

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 *)

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

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).