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A157521
Triangle T(n, k) = n! * binomial(n, k)*( psi(n-k+1) - psi(k+1) ), read by rows.
2
0, 1, -1, 3, 0, -3, 11, 9, -9, -11, 50, 80, 0, -80, -50, 274, 650, 400, -400, -650, -274, 1764, 5544, 6300, 0, -6300, -5544, -1764, 13068, 51156, 82908, 44100, -44100, -82908, -51156, -13068, 109584, 513792, 1072512, 1016064, 0, -1016064, -1072512, -513792, -109584
OFFSET
0,4
FORMULA
T(n, k) = n! * (d/dk) binomial(n, k).
Sum_{k=0..n} T(n, k) = 0.
From G. C. Greubel, Jan 13 2022: (Start)
T(n, k) = n! * binomial(n, k)*( psi(n-k+1) - psi(k+1) ), psi = digamma function.
T(2*n, n) = 0.
T(n, n-k) = - T(n, k), k <= floor(n/2).
T(n, 0) = A000254(n). (End)
EXAMPLE
Triangle begins as:
0;
1, -1;
3, 0, -3;
11, 9, -9, -11;
50, 80, 0, -80, -50;
274, 650, 400, -400, -650, -274;
1764, 5544, 6300, 0, -6300, -5544, -1764;
13068, 51156, 82908, 44100, -44100, -82908, -51156, -13068;
109584, 513792, 1072512, 1016064, 0, -1016064, -1072512, -513792, -109584;
MATHEMATICA
T[n_, k_]:= n!*Binomial[n, k]*(PolyGamma[0, n-k+1] - PolyGamma[0, k+1]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 13 2022 *)
PROG
(Magma) [Round(Factorial(n)*Binomial(n, k)*(Psi(n-k+1) - Psi(k+1))): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 13 2022
(Sage) flatten([[factorial(n)*binomial(n, k)*(psi(n-k+1) - psi(k+1)) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jan 13 2022
CROSSREFS
Sequence in context: A191619 A327245 A157525 * A176005 A211963 A275080
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Mar 02 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 13 2022
STATUS
approved