A047668 -id:A047668 - cp's OEIS frontend

A047666 Square array a(n,k) read by antidiagonals: a(n,1)=n, a(1,k)=k, a(n,k) = a(n-1,k-1) + a(n-1,k) + a(n,k-1).

1, 2, 2, 3, 5, 3, 4, 10, 10, 4, 5, 17, 25, 17, 5, 6, 26, 52, 52, 26, 6, 7, 37, 95, 129, 95, 37, 7, 8, 50, 158, 276, 276, 158, 50, 8, 9, 65, 245, 529, 681, 529, 245, 65, 9, 10, 82, 360, 932, 1486, 1486, 932, 360, 82, 10, 11, 101, 507, 1537, 2947, 3653

Offset: 1

N. J. A. Sloane

Main diagonal is A002002. Rows give A002522, A047667, A047668, ...

A047666 := proc(n,k) option remember; if n = 1 then k; elif k = 1 then n; else A047666(n-1,k-1)+A047666(n,k-1)+A047666(n-1,k); fi; end;

nmax = 11; a[1, k_] := k; a[n_, 1] := n; a[n_, k_] := a[n, k] = a[n-1, k-1] + a[n, k-1] + a[n-1, k]; Flatten[ Table[ a[n-k+1, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Feb 10 2012 *)

T(n, m) = (Sum_{i=1..n-m}(2*i+1)*U(n-i-1, m-1)) + (Sum_{i=1..m} (2*i+1)*U(n-2, m-i)) - U(n-2, m-1) where U(n, m) = A008288(n, m). - Floor van Lamoen, Aug 16 2001

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

Original entry on oeis.org

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

Wolfdieter Lang, Jun 11 2004

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]

Partial sums of A060488.

A095667:=n->(n+16)*binomial(n+3,3)/4; seq(A095667(k), k=0..70); # Wesley Ivan Hurt, Sep 25 2013

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 *)

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

cp's OEIS Frontend

A047666 Square array a(n,k) read by antidiagonals: a(n,1)=n, a(1,k)=k, a(n,k) = a(n-1,k-1) + a(n-1,k) + a(n,k-1).

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