A222061 - OEIS

%I #14 Jan 01 2018 04:51:33

%S 0,0,1,0,1,5,0,1,15,26,0,1,35,156,154,0,1,75,650,1540,1044,0,1,155,

%T 2340,10010,15660,8028,0,1,315,7826,53900,146160,168588,69264,0,1,635,

%U 25116,261954,1096200,2135448,1939392,663696,0,1,1275,78650,1196580,7256844

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

%H Ayhan Dil and Veli Kurt, <a href="http://www.emis.de/journals/INTEGERS/papers/m38/m38.Abstract.html">Polynomials related to harmonic numbers and evaluation of harmonic number series I</a>, INTEGERS, 12 (2012), #A38.

%F 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

%e Triangle begins:

%e   0

%e   0 1

%e   0 1 5

%e   0 1 15 26

%e   0 1 35 156 154

%e   0 1 75 650 1540 1044

%e   ....

%t H2[k_] := (k+1) (HarmonicNumber[k+1] - 1);

%t T[n_, k_] := StirlingS2[n, k] k! H2[k];

%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 01 2018 *)

%o (PARI)

%o hyp(n,alpha) = {x= y+O(y^(n+1)); gf = - log(1-x)/(1-x)^alpha; polcoeff(gf, n, y);}

%o t(n, k) = {k!*(sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!)*hyp(k,2)};

%o \\ _Michel Marcus_, Feb 09 2013

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

%K nonn,tabl

%O 0,6

%A _N. J. A. Sloane_, Feb 08 2013

%E More terms from _Michel Marcus_, Feb 09 2013