A095667 Fifth column (m=4) of (1,4)-Pascal triangle A095666.

4, 17, 45, 95, 175, 294, 462, 690, 990, 1375, 1859, 2457, 3185, 4060, 5100, 6324, 7752, 9405, 11305, 13475, 15939, 18722, 21850, 25350, 29250, 33579, 38367, 43645, 49445, 55800, 62744, 70312, 78540, 87465, 97125, 107559, 118807, 130910, 143910, 157850

Offset: 0

Author

Wolfdieter Lang, Jun 11 2004

Keywords

• nonn
• easy

Comments

If Y is a 4-subset of an n-set X then, for n>=7, a(n-7) is the number of 4-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007

In this sequence if we do a forward difference, then the 3rd forward difference when considered as a sequence will be an arithmetic progression with common difference 1. The same way the sequence formed by the 3rd forward difference of A047668 will be an arithmetic progression with common difference 8. [From Gopalakrishnan (gopala498(AT)yahoo.co.in), Jun 05 2010]

Row 4 of the convolution array A213550. [Clark Kimberling, Jun 20 2012]

Crossrefs

Partial sums of A060488.

Programs

Maple

A095667:=n->(n+16)*binomial(n+3,3)/4; seq(A095667(k), k=0..70); # Wesley Ivan Hurt, Sep 25 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,3,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 04 2009 *)

Formula

G.f.: (4-3*x)/(1-x)^5.

a(n) = 4*b(n)-3*b(n-1) = (n+16)*binomial(n+3, 3)/4, with b(n):=binomial(n+4, 4)= A000332(n+4, 4).

a(n) = sum_{k=1..n} ( sum_{i=1..k} i*(n-k+4) ). - Wesley Ivan Hurt, Sep 25 2013