A095661 Fifth column (m=4) of (1,3)-Pascal triangle A095660.

3, 13, 35, 75, 140, 238, 378, 570, 825, 1155, 1573, 2093, 2730, 3500, 4420, 5508, 6783, 8265, 9975, 11935, 14168, 16698, 19550, 22750, 26325, 30303, 34713, 39585, 44950, 50840, 57288, 64328, 71995, 80325, 89355, 99123, 109668, 121030, 133250, 146370

Offset: 0

Author

Wolfdieter Lang, Jun 11 2004

Keywords

• nonn
• easy

Comments

If Y is a 3-subset of an n-set X then, for n>=6, a(n-6) is the number of 4-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

Row 3 of the convolution array A213550. - Clark Kimberling, Jun 20 2012

Crossrefs

Partial sums of A006503.

Cf. A213550.

Programs

Maple

A095661:=n->(n+12)*binomial(n+3, 3)/4; seq(A095661(k), k=0..50); # Wesley Ivan Hurt, Oct 10 2013

Mathematica

s1=s2=s3=s4=0;lst={};Do[a=n+(n+2);s1+=a;s2+=s1;s3+=s2;s4+=s3;AppendTo[lst,s3/2],{n,2,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 04 2009 *)
Table[(n+12)Binomial[n+3, 3]/4, {n, 0, 50}] (* Wesley Ivan Hurt, Oct 10 2013 *)

Formula

G.f.: (3-2*x)/(1-x)^5.

a(n) = (n+12)*binomial(n+3, 3)/4 = 3*b(n)-2*b(n-1), with b(n) := binomial(n+4, 4); cf. A000332.

a(n) = Sum_{k=1..n} Sum_{i=1..k} i*(n-k+3), with offset 1. - Wesley Ivan Hurt, Sep 25 2013