A081130 Square array of binomial transforms of (0,0,1,0,0,0,...), read by antidiagonals.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 6, 6, 0, 0, 0, 1, 9, 24, 10, 0, 0, 0, 1, 12, 54, 80, 15, 0, 0, 0, 1, 15, 96, 270, 240, 21, 0, 0, 0, 1, 18, 150, 640, 1215, 672, 28, 0, 0, 0, 1, 21, 216, 1250, 3840, 5103, 1792, 36, 0, 0, 0, 1, 24, 294, 2160, 9375, 21504, 20412, 4608, 45, 0

Offset

0,14

Comments

Rows, of the square array, are three-fold convolutions of sequences of powers.

Links

G. C. Greubel, Antidiadoganal rows n = 0..50, flattened

Formula

T(n, k) = k^(n-2)*binomial(n, 2), with T(n, 0) = 0 (square array).

T(n, n) = A081131(n).

Rows have g.f. x^3/(1-k*x)^n.

From G. C. Greubel, May 14 2021: (Start)

T(k, n-k) = (n-k)^(k-2)*binomial(k,2) with T(n, n) = 0 (antidiagonal triangle).

Sum_{k=0..n} T(n, n-k) = A081197(n). (End)

Example

The array begins as:
  0,  0,  0,   0,   0,    0, ...
  0,  0,  0,   0,   0,    0, ...
  0,  1,  1,   1,   1,    1, ... A000012
  0,  3,  6,   9,  12,   15, ... A008585
  0,  6, 24,  54,  96,  150, ... A033581
  0, 10, 80, 270, 640, 1250, ... A244729
The antidiagonal triangle begins as:
  0;
  0, 0;
  0, 0, 0;
  0, 0, 1, 0;
  0, 0, 1, 3,  0;
  0, 0, 1, 6,  6,  0;
  0, 0, 1, 9, 24, 10, 0;

Mathematica

Table[If[k==n, 0, (n-k)^(k-2)*Binomial[k, 2]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 14 2021 *)

Prog

(Magma) [k eq n select 0 else (n-k)^(k-2)*Binomial(k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, May 14 2021
(Sage) flatten([[0 if (k==n) else (n-k)^(k-2)*binomial(k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 14 2021
(PARI)
T(n, k)=if (k==0, 0, k^(n-2)*binomial(n, 2));
seq(nn) = for (n=0, nn, for (k=0, n, print1(T(k, n-k), ", ")); );
seq(12) \\ Michel Marcus, May 14 2021

Crossrefs

Main diagonal: A081131.

Rows: A000012 (n=2), A008585 (n=3), A033581 (n=4), A244729 (n=5).

Columns: A000217 (k=1), A001788 (k=2), A027472 (k=3), A038845 (k=4), A081135 (k=5), A081136 (k=6), A027474 (k=7), A081138 (k=8), A081139 (k=9), A081140 (k=10), A081141 (k=11), A081142 (k=12), A027476 (k=15).

Sequence in context: A035641 A242434 A036873 * A358623 A360224 A174428

Adjacent sequences: A081127 A081128 A081129 * A081131 A081132 A081133

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 08 2003

Extensions

Term a(5) corrected by G. C. Greubel, May 14 2021

Status

approved