A081130 - OEIS

%I #18 Sep 08 2022 08:45:09

%S 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,

%T 54,80,15,0,0,0,1,15,96,270,240,21,0,0,0,1,18,150,640,1215,672,28,0,0,

%U 0,1,21,216,1250,3840,5103,1792,36,0,0,0,1,24,294,2160,9375,21504,20412,4608,45,0

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

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

%H G. C. Greubel, <a href="/A081130/b081130.txt">Antidiadoganal rows n = 0..50, flattened</a>

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

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

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

%F From _G. C. Greubel_, May 14 2021: (Start)

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

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

%e The array begins as:

%e   0,  0,  0,   0,   0,    0, ...

%e   0,  0,  0,   0,   0,    0, ...

%e   0,  1,  1,   1,   1,    1, ... A000012

%e   0,  3,  6,   9,  12,   15, ... A008585

%e   0,  6, 24,  54,  96,  150, ... A033581

%e   0, 10, 80, 270, 640, 1250, ... A244729

%e The antidiagonal triangle begins as:

%e   0;

%e   0, 0;

%e   0, 0, 0;

%e   0, 0, 1, 0;

%e   0, 0, 1, 3,  0;

%e   0, 0, 1, 6,  6,  0;

%e   0, 0, 1, 9, 24, 10, 0;

%t 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 *)

%o (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

%o (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

%o (PARI)

%o T(n, k)=if (k==0, 0, k^(n-2)*binomial(n, 2));

%o seq(nn) = for (n=0, nn, for (k=0, n, print1(T(k, n-k), ", ")); );

%o seq(12) \\ _Michel Marcus_, May 14 2021

%Y Main diagonal: A081131.

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

%Y 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).

%K easy,nonn,tabl

%O 0,14

%A _Paul Barry_, Mar 08 2003

%E Term a(5) corrected by _G. C. Greubel_, May 14 2021