A247608 a(n) = Sum_{k=0..3} binomial(6,k)*binomial(n,k).

1, 7, 28, 84, 195, 381, 662, 1058, 1589, 2275, 3136, 4192, 5463, 6969, 8730, 10766, 13097, 15743, 18724, 22060, 25771, 29877, 34398, 39354, 44765, 50651, 57032, 63928, 71359, 79345, 87906, 97062, 106833, 117239, 128300, 140036, 152467, 165613, 179494

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 (4,-6,4,-1).

Crossrefs

Cf. A005408, A056108.

Programs

Magma

[(6+31*n-15*n^2+20*n^3)/6: n in [0..40]];

Magma

[1+6*Binomial(n,1)+15*Binomial(n,2)+20*Binomial(n,3): n in [0..40]];

Magma

I:=[1, 7, 28, 84]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]];

Mathematica

Table[(6 + 31 n - 15 n^2 + 20 n^3)/6, {n, 0, 50}] (* or *) CoefficientList[Series[(1 + 3 x + 6 x^2 + 10 x^3)/(1-x)^4,{x, 0, 50}], x]

PARI

Vec((1+3*x+6*x^2+10*x^3)/(1-x)^4 + O (x^50)) \\ Michel Marcus, Sep 22 2014

Sage

m=3; [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+3*x+6*x^2+10*x^3)/(1-x)^4.

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

a(n) = (6+31*n-15*n^2+20*n^3)/6.

a(n) = 1+6*Binomial(n,1)+15*Binomial(n,2)+20*Binomial(n,3).