A062966 a(n) = C(3+n, n) + C(4+n, n) + C(5+n, n) + C(6+n, n).

4, 22, 74, 195, 441, 896, 1680, 2958, 4950, 7942, 12298, 18473, 27027, 38640, 54128, 74460, 100776, 134406, 176890, 229999, 295757, 376464, 474720, 593450, 735930, 905814, 1107162, 1344469, 1622695, 1947296

Offset: 0

Author

Wolfdieter Lang, Jul 12 2001

Keywords

• nonn
• easy

Comments

In the Frey-Sellers reference this sequence is called {(n+2) over 6}_{3}, n >= 0.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
• Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Crossrefs

Seventh column (r=6) of FS(4) staircase array A062750.

Partial sums of A027659.

Programs

Mathematica

Table[Sum[Binomial[i+n,n],{i,3,6}],{n,0,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{4,22,74,195,441,896,1680},30] (* Harvey P. Dale, May 02 2012 *)

PARI

{ for (n=0, 1000, a=binomial(3 + n, n) + binomial(4 + n, n) + binomial(5 + n, n) + binomial(6 + n, n); write("b062966.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 14 2009

Formula

a(n) = A062750(n+2, 6) = (n+10)*(n+3)*(n+2)*(n+1)*(n^2+11*n+48)/6!.

G.f.: = (x-2)*(x^2-2*x+2)/(x-1)^7 = N(4;1, x)/(1-x)^7 with N(4;1, x) = 4-6*x+4*x^2-x^3, polynomial of second row of A062751.

a(0)=4, a(1)=22, a(2)=74, a(3)=195, a(4)=441, a(5)=896, a(6)=1680, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Harvey P. Dale, May 02 2012

Extensions

Better description from Zerinvary Lajos, Dec 02 2005