A116923 Triangle T(n,k) = Sum_{i=0..k} (-1)^(i+k)*binomial(k,i)*Sum_{j=0..n} (i+1)^j*(3n-3j+1) read by rows.

1, 5, 1, 12, 7, 2, 22, 26, 20, 6, 35, 74, 112, 84, 24, 51, 183, 484, 672, 456, 120, 70, 417, 1818, 4140, 4968, 3000, 720, 92, 904, 6288, 22014, 41400, 42840, 23040, 5040, 117, 1900, 20672, 106920, 295056, 464040, 418320, 201600, 40320, 145, 3917, 65816, 489696, 1902960

Offset

0,2

Comments

Start from an array A(n,i) = Sum_{j=0..n} (i+1)^j*(3n-3j+1) which contains A000012

in row n=0, A000027 starting from 5 in row n=1, A117950 starting from 12 in row n=2 etc.

T(n,.) is obtained by computing the inverse binomial transform of row n of A(.,.).

Since A(n,i) is a polynomial in i, the inverse binomial transform is essentially finite; all trailing zeros are discarded while building the triangle.

Links

Table of n, a(n) for n=0..49.

Formula

T(n,0) = A000326(n+1).

T(n,n) = n! = A000142(n).

T(n,k) = Sum_{j=0..n} A028246(j+1,k+1)*(3*n-3*j+1). - R. J. Mathar, Mar 27 2010

Example

First few rows of the array A(.,.):
   1,  1,  1,  1,  1, ...
   5,  6,  7,  8,  9, ...
  12, 19, 28, 39, 52, ...
  ...
such that for example the inverse binomial transform of 12, 19, 28, ... becomes row n=2 of the triangle: 12, 7, 2, (0, 0, 0, 0, ...).
First few rows of the triangle T(n,k):
   1,
   5,   1;
  12,   7,   2;
  22,  26,  20,   6;
  35,  74, 112,  84,  24;
  51, 183, 484, 672, 456, 120;
  ...

Maple

A := proc(n, i) add( (i+1)^j*(3*n-3*j+1), j=0..n) ; end proc:
A116923 := proc(n, m) add((-1)^(i+m)*binomial(m, i)*A(n, i), i=0..m) ; end proc:
seq(seq(A116923(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Mar 27 2010

Crossrefs

Cf. A000326.

Sequence in context: A343223 A104572 A125232 * A327797 A062264 A276738

Adjacent sequences: A116920 A116921 A116922 * A116924 A116925 A116926

Keyword

nonn,tabl

Author

Gary W. Adamson, Feb 26 2006

Extensions

Offset set to 0 and precise definition added by R. J. Mathar, Mar 27 2010

Status

approved