%I #11 Aug 03 2021 01:52:47
%S 1,2,4,6,9,9,24,56,24,16,120,250,275,50,25,720,1884,1350,960,90,36,
%T 5040,12348,14896,5145,2695,147,49,40320,114624,105056,80416,15680,
%U 6496,224,64,362880,986256,1282284,605556,336609,40824,13986,324,81
%N Triangle T(n, k) = coefficients of (n+1)!*(binomial(x+n+1, n+1) - binomial(x, n+1)), read by rows.
%D Brendan Hassett, Introduction to algebraic Geometry, Cambridge University Press, New York, 2007, page 214
%H G. C. Greubel, <a href="/A178126/b178126.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = coefficients of n!*(binomial(x+n+1, n+1) - binomial(x, n+1)).
%F From _G. C. Greubel_, Apr 14 2021: (Start)
%F T(n, k) = coefficients of Sum_{j=0..n+1} Stirling1(n+1, j)*( (x+n+1)^j - x^j ).
%F T(n, 0) = (n+1)!.
%F T(n, n) = (n+1)^2.
%F Sum_{k=0..n} T(n,k) = (n+2)! - [n=0]. (End)
%e Triangle begins as:
%e 1;
%e 2, 4;
%e 6, 9, 9;
%e 24, 56, 24, 16;
%e 120, 250, 275, 50, 25;
%e 720, 1884, 1350, 960, 90, 36;
%e 5040, 12348, 14896, 5145, 2695, 147, 49;
%e 40320, 114624, 105056, 80416, 15680, 6496, 224, 64;
%e 362880, 986256, 1282284, 605556, 336609, 40824, 13986, 324, 81;
%e 3628800, 10991520, 11727000, 9582200, 2693250, 1171380, 94500, 27600, 450, 100;
%t T[n_, k_]:= SeriesCoefficient[Series[Sum[StirlingS1[n+1, j]*((x+n+1)^j -x^j), {j, 0, n+1}], {x, 0, n+1}], k];
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Apr 14 2021 *)
%o (Sage)
%o def T(n,k): return ( sum((-1)^(n+j+1)*stirling_number1(n+1, j)*((x+n+1)^j - x^j) for j in (0..n+1)) ).series(x,n+1).list()[k]
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 14 2021
%Y Cf. A048994, A139167.
%K nonn,tabl
%O 0,2
%A _Roger L. Bagula_, May 20 2010
%E Edited by _G. C. Greubel_, Apr 14 2021