A222061 Triangle read by rows: coefficients of second-order hypergeometric-harmonic polynomials.

0, 0, 1, 0, 1, 5, 0, 1, 15, 26, 0, 1, 35, 156, 154, 0, 1, 75, 650, 1540, 1044, 0, 1, 155, 2340, 10010, 15660, 8028, 0, 1, 315, 7826, 53900, 146160, 168588, 69264, 0, 1, 635, 25116, 261954, 1096200, 2135448, 1939392, 663696, 0, 1, 1275, 78650, 1196580, 7256844

Offset: 0

Author

N. J. A. Sloane, Feb 08 2013

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  0
  0 1
  0 1 5
  0 1 15 26
  0 1 35 156 154
  0 1 75 650 1540 1044
  ....
		

Links

• Ayhan Dil and Veli Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I, INTEGERS, 12 (2012), #A38.

Crossrefs

Cf. A222057-A222064. Row sums are in A222062.

Programs

Mathematica

H2[k_] := (k+1) (HarmonicNumber[k+1] - 1);
T[n_, k_] := StirlingS2[n, k] k! H2[k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 01 2018 *)

PARI

hyp(n,alpha) = {x= y+O(y^(n+1)); gf = - log(1-x)/(1-x)^alpha; polcoeff(gf, n, y);}
t(n, k) = {k!*(sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!)*hyp(k,2)};
\\ Michel Marcus, Feb 09 2013

Formula

T(n,k) = A008277(n,k)*A000142(k)*H2(k) where H2(k) is defined by g.f.:- log(1-x)/(1-x)^2. - Michel Marcus, Feb 09 2013

Extensions

More terms from Michel Marcus, Feb 09 2013