OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
Peter Luschny, A sequence transformation and the Bernoulli numbers.
FORMULA
T(n, k) = abs(Stirling1(k+1, 2) * Stirling2(n+1, k+1)).
EXAMPLE
Triangle begins as:
0;
0, 1;
0, 3, 3;
0, 7, 18, 11;
0, 15, 75, 110, 50;
0, 31, 270, 715, 750, 274;
MAPLE
T176276 := proc(n, k) local W, H;
W := proc(n, k) stirling2(n+1, k+1)*k! end:
H := proc(n) local i; add(1/i, i=1..n) end: # H(0) = 0 (empty sum convention)
W(n, k)*H(k) end:
MATHEMATICA
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 *)
PROG
(PARI) T(n, k) = k!*stirling(n+1, k+1, 2)*sum(j=1, k, 1/j); \\ G. C. Greubel, Nov 24 2019
(Magma) [Abs(StirlingFirst(k+1, 2)*StirlingSecond(n+1, k+1)): k in [0..n], n in [0..10]];
(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
(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
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Apr 14 2010
STATUS
approved