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

%I A222061 #14 Jan 01 2018 04:51:33
%S A222061 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 A222061 2340,10010,15660,8028,0,1,315,7826,53900,146160,168588,69264,0,1,635,
%U A222061 25116,261954,1096200,2135448,1939392,663696,0,1,1275,78650,1196580,7256844
%N A222061 Triangle read by rows: coefficients of second-order hypergeometric-harmonic polynomials.
%H A222061 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 A222061 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 A222061 Triangle begins:
%e A222061   0
%e A222061   0 1
%e A222061   0 1 5
%e A222061   0 1 15 26
%e A222061   0 1 35 156 154
%e A222061   0 1 75 650 1540 1044
%e A222061   ....
%t A222061 H2[k_] := (k+1) (HarmonicNumber[k+1] - 1);
%t A222061 T[n_, k_] := StirlingS2[n, k] k! H2[k];
%t A222061 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 01 2018 *)
%o A222061 (PARI)
%o A222061 hyp(n,alpha) = {x= y+O(y^(n+1)); gf = - log(1-x)/(1-x)^alpha; polcoeff(gf, n, y);}
%o A222061 t(n, k) = {k!*(sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!)*hyp(k,2)};
%o A222061 \\ _Michel Marcus_, Feb 09 2013
%Y A222061 Cf. A222057-A222064. Row sums are in A222062.
%K A222061 nonn,tabl
%O A222061 0,6
%A A222061 _N. J. A. Sloane_, Feb 08 2013
%E A222061 More terms from _Michel Marcus_, Feb 09 2013