A178126 Triangle T(n, k) = coefficients of (n+1)!*(binomial(x+n+1, n+1) - binomial(x, n+1)), read by rows.
1, 2, 4, 6, 9, 9, 24, 56, 24, 16, 120, 250, 275, 50, 25, 720, 1884, 1350, 960, 90, 36, 5040, 12348, 14896, 5145, 2695, 147, 49, 40320, 114624, 105056, 80416, 15680, 6496, 224, 64, 362880, 986256, 1282284, 605556, 336609, 40824, 13986, 324, 81
Offset: 0
Examples
Triangle begins as:
1;
2, 4;
6, 9, 9;
24, 56, 24, 16;
120, 250, 275, 50, 25;
720, 1884, 1350, 960, 90, 36;
5040, 12348, 14896, 5145, 2695, 147, 49;
40320, 114624, 105056, 80416, 15680, 6496, 224, 64;
362880, 986256, 1282284, 605556, 336609, 40824, 13986, 324, 81;
3628800, 10991520, 11727000, 9582200, 2693250, 1171380, 94500, 27600, 450, 100;
References
- Brendan Hassett, Introduction to algebraic Geometry, Cambridge University Press, New York, 2007, page 214
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Mathematica
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]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 14 2021 *) -
Sage
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] flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 14 2021
Formula
T(n, k) = coefficients of n!*(binomial(x+n+1, n+1) - binomial(x, n+1)).
From G. C. Greubel, Apr 14 2021: (Start)
T(n, k) = coefficients of Sum_{j=0..n+1} Stirling1(n+1, j)*( (x+n+1)^j - x^j ).
T(n, 0) = (n+1)!.
T(n, n) = (n+1)^2.
Sum_{k=0..n} T(n,k) = (n+2)! - [n=0]. (End)
Extensions
Edited by G. C. Greubel, Apr 14 2021