OFFSET
0,5
COMMENTS
Row sums are: {1, 2, 52, 2524, 321188, 55440032, 17484363762, 7778873938968, 5190384632566660, 4688678305303834312, ...}.
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
T(n, m) = 2 + binomial(n, m)^3 - 3*binomial(n, m)*Eulerian(n+1, m) + Eulerian(n+1, m)^3, where Eulerian(n,k) = A008292(n,k).
EXAMPLE
1;
1, 1;
1, 50, 1;
1, 1261, 1261, 1;
1, 17330, 286526, 17330, 1;
1, 184465, 27535550, 27535550, 184465, 1;
1, 1726058, 1689360653, 14102190338, 1689360653, 1726058, 1;
MATHEMATICA
Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];
T[n_, m_]:= 2 + Binomial[n, m]^3 - 3*Binomial[n, m]*Eulerian[n+1, m] + Eulerian[n+1, m]^3;
Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 26 2019 *)
PROG
(PARI) Eulerian(n, k) = sum(j=0, k+1, (-1)^j*binomial(n+1, j)*(k-j+1)^n);
{T(n, k) = 2 + binomial(n, k)^3 - 3*binomial(n, k)*Eulerian(n+1, k) + Eulerian(n+1, k)^3 };
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Apr 26 2019
(Magma) Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
[[2 +Binomial(n, k)^3 -3*Binomial(n, k)*Eulerian(n+1, k) +Eulerian(n+1, k)^3: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Apr 26 2019
(Sage)
def Eulerian(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in (0..k+1))
def T(n, k): return 2 + binomial(n, k)^3 -3*binomial(n, k)*Eulerian(n+1, k) + Eulerian(n+1, k)^3
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Apr 26 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Apr 02 2010
EXTENSIONS
Edited by G. C. Greubel, Apr 26 2019
STATUS
approved