A174914 A symmetrical triangle sequence T(n, m) = 2 + binomial(n, m)^3 - 3*binomial(n, m)*Eulerian(n+1, m) + Eulerian(n+1, m)^3.

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, 1, 15064381, 79119080561, 3810302824541, 3810302824541, 79119080561, 15064381, 1

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

Cf. A008292.

Sequence in context: A015066 A022078 A337330 * A036172 A242075 A187526

Adjacent sequences: A174911 A174912 A174913 * A174915 A174916 A174917

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 02 2010

Extensions

Edited by G. C. Greubel, Apr 26 2019

Status

approved