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
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.
Programs
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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