A247610 a(n) = Sum_{k=0..5} binomial(10,k)*binomial(n,k).

1, 11, 66, 286, 1001, 3003, 7798, 17858, 36873, 70003, 124130, 208110, 333025, 512435, 762630, 1102882, 1555697, 2147067, 2906722, 3868382, 5070009, 6554059, 8367734, 10563234, 13198009, 16335011, 20042946, 24396526, 29476721, 35371011, 42173638, 49985858

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 (6,-15,20,-15,6,-1).

Crossrefs

Cf. A005408, A056108, A247608.

Programs

Magma

[(20+508*n-925*n^2+820*n^3-245*n^4+42*n^5)/20: n in [0..40]];

Magma

I:=[1,11,66,286,1001,3003]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]];

Mathematica

Table[(20 + 508 n - 925 n^2 + 820 n^3 - 245 n^4 + 42 n^5)/20, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 5 x + 15 x^2 + 35 x^3 + 70 x^4 + 126 x^5)/(1 - x)^6, {x, 0, 40}], x]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,11,66,286,1001,3003},40] (* Harvey P. Dale, Apr 20 2022 *)

Sage

m=5; [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 + 5*x + 15*x^2 + 35*x^3 + 70*x^4 + 126*x^5) / (1-x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

a(n) = (20 + 508*n - 925*n^2 + 820*n^3 - 245*n^4 + 42*n^5)/20.