A157525 Triangle T(n, k) = n! * (Harmonic number(n-k) - Harmonic number(k)), read by rows.

0, 1, -1, 3, 0, -3, 11, 3, -3, -11, 50, 20, 0, -20, -50, 274, 130, 40, -40, -130, -274, 1764, 924, 420, 0, -420, -924, -1764, 13068, 7308, 3948, 1260, -1260, -3948, -7308, -13068, 109584, 64224, 38304, 18144, 0, -18144, -38304, -64224, -109584, 1026576, 623376, 396576, 223776, 72576, -72576, -223776, -396576, -623376, -1026576

Offset

0,4

Links

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

Formula

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

T(n, 0) = A000254(n).

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

From G. C. Greubel, Jan 23 2022: (Start)

T(n, k) = n! * (psi(n-k+1) - psi(k+1)), where psi(x) = digamma(x).

T(n, k) = n! * (H(n-k) - H(k)), where H(n) = harmonic number(n).

T(n, n-k) = -T(n, k).

T(2*n, n) = 0. (End)

Example

Triangle begins as:
       0;
       1,    -1;
       3,     0,    -3;
      11,     3,    -3,   -11;
      50,    20,     0,   -20,   -50;
     274,   130,    40,   -40,  -130,   -274;
    1764,   924,   420,     0,  -420,   -924,  -1764;
   13068,  7308,  3948,  1260, -1260,  -3948,  -7308, -13068;
  109584, 64224, 38304, 18144,     0, -18144, -38304, -64224, -109584;

Mathematica

T[n_, k_]:= n!*(PolyGamma[0, n-k+1] - PolyGamma[0, k+1]);
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 23 2022 *)

Prog

(Magma)
T:= func< n, k | Round(Factorial(n)*(Psi(n-k+1) - Psi(k+1))) >;
[T(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 23 2022
(Sage)
def T(n, k): return factorial(n)*(harmonic_number(n-k) - harmonic_number(k))
flatten([[T(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jan 23 2022

Crossrefs

Cf. A000254, A157521.

Sequence in context: A212036 A191619 A327245 * A157521 A176005 A211963

Adjacent sequences: A157522 A157523 A157524 * A157526 A157527 A157528

Keyword

sign,tabl

Author

Roger L. Bagula, Mar 02 2009

Extensions

Edited by G. C. Greubel, Jan 23 2022

Status

approved