A157521 Triangle T(n, k) = n! * binomial(n, k)*( psi(n-k+1) - psi(k+1) ), read by rows.

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

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Cf. A000254, A157525.

Sequence in context: A191619 A327245 A157525 * A176005 A211963 A275080

Adjacent sequences: A157518 A157519 A157520 * A157522 A157523 A157524

Keyword

sign,tabl

Author

Roger L. Bagula, Mar 02 2009

Extensions

Edited by G. C. Greubel, Jan 13 2022

Status

approved