A157521 - OEIS

%I #9 Sep 08 2022 08:45:42

%S 0,1,-1,3,0,-3,11,9,-9,-11,50,80,0,-80,-50,274,650,400,-400,-650,-274,

%T 1764,5544,6300,0,-6300,-5544,-1764,13068,51156,82908,44100,-44100,

%U -82908,-51156,-13068,109584,513792,1072512,1016064,0,-1016064,-1072512,-513792,-109584

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

%H G. C. Greubel, <a href="/A157521/b157521.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = n! * (d/dk) binomial(n, k).

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

%F From _G. C. Greubel_, Jan 13 2022: (Start)

%F T(n, k) = n! * binomial(n, k)*( psi(n-k+1) - psi(k+1) ), psi = digamma function.

%F T(2*n, n) = 0.

%F T(n, n-k) = - T(n, k), k <= floor(n/2).

%F T(n, 0) = A000254(n). (End)

%e Triangle begins as:

%e        0;

%e        1,     -1;

%e        3,      0,      -3;

%e       11,      9,      -9,     -11;

%e       50,     80,       0,     -80,    -50;

%e      274,    650,     400,    -400,   -650,     -274;

%e     1764,   5544,    6300,       0,  -6300,    -5544,    -1764;

%e    13068,  51156,   82908,   44100, -44100,   -82908,   -51156,  -13068;

%e   109584, 513792, 1072512, 1016064,      0, -1016064, -1072512, -513792, -109584;

%t T[n_, k_]:= n!*Binomial[n, k]*(PolyGamma[0, n-k+1] - PolyGamma[0, k+1]);

%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 13 2022 *)

%o (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

%o (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

%Y Cf. A000254, A157525.

%K sign,tabl

%O 0,4

%A _Roger L. Bagula_, Mar 02 2009

%E Edited by _G. C. Greubel_, Jan 13 2022