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%I #13 Sep 08 2022 08:45:52
%S 0,0,1,0,3,3,0,7,18,11,0,15,75,110,50,0,31,270,715,750,274,0,63,903,
%T 3850,7000,5754,1764,0,127,2898,18711,52500,72884,49392,13068,0,255,
%U 9075,85470,347550,725004,814968,470448,109584,0,511,27990,375155,2126250,6254598,10372320,9801000,4931280,1026576
%N Worpitzky(n, k)*Harmonic(k), triangle read by rows.
%H G. C. Greubel, <a href="/A176276/b176276.txt">Rows n = 0..100 of triangle, flattened</a>
%H Peter Luschny, <a href="http://www.oeis.org/wiki/User:Peter_Luschny/SeqTransformation">A sequence transformation and the Bernoulli numbers</a>.
%F T(n, k) = abs(Stirling1(k+1, 2) * Stirling2(n+1, k+1)).
%e Triangle begins as:
%e 0;
%e 0, 1;
%e 0, 3, 3;
%e 0, 7, 18, 11;
%e 0, 15, 75, 110, 50;
%e 0, 31, 270, 715, 750, 274;
%p T176276 := proc(n, k) local W, H;
%p W := proc(n, k) stirling2(n+1, k+1)*k! end:
%p H := proc(n) local i; add(1/i, i=1..n) end: # H(0) = 0 (empty sum convention)
%p W(n, k)*H(k) end:
%t T[n_, k_]:= StirlingS2[n+1, k+1]*k!*HarmonicNumber[k]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Jul 29 2013 *)
%o (PARI) T(n,k) = k!*stirling(n+1,k+1,2)*sum(j=1,k,1/j); \\ _G. C. Greubel_, Nov 24 2019
%o (Magma) [Abs(StirlingFirst(k+1, 2)*StirlingSecond(n+1, k+1)): k in [0..n], n in [0..10]];
%o (Sage) [[factorial(k)*stirling_number1(n+1,k+1)*harmonic_number(k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 24 2019
%o (GAP) Flat(List([0..10], n-> List([0..n], k-> AbsInt(Stirling1(k+1, 2) * Stirling2(n+1, k+1)) ))); # _G. C. Greubel_, Nov 24 2019
%Y Cf. A028246, A176277.
%K easy,nonn,tabl
%O 0,5
%A _Peter Luschny_, Apr 14 2010
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