A176200 A symmetrical triangle T(n, m) = 2*Eulerian(n+1, m) -1, read by rows.

1, 1, 1, 1, 7, 1, 1, 21, 21, 1, 1, 51, 131, 51, 1, 1, 113, 603, 603, 113, 1, 1, 239, 2381, 4831, 2381, 239, 1, 1, 493, 8585, 31237, 31237, 8585, 493, 1, 1, 1003, 29215, 176467, 312379, 176467, 29215, 1003, 1, 1, 2025, 95679, 910383, 2620707, 2620707, 910383, 95679, 2025, 1

Offset

0,5

Comments

Row sums are: {1, 2, 9, 44, 235, 1434, 10073, 80632, 725751, 7257590, 79833589, ...}.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n, m) = 2*Eulerian(n+1, m) - 1, where Eulerian(n, k) = A008292(n,k).

Example

Triangle begins as:
  1;
  1,   1;
  1,   7,    1;
  1,  21,   21,     1;
  1,  51,  131,    51,     1;
  1, 113,  603,   603,   113,    1;
  1, 239, 2381,  4831,  2381,  239,   1;
  1, 493, 8585, 31237, 31237, 8585, 493, 1;

Mathematica

Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];
T[n_, m_]:= 2*Eulerian[n+1, m]-1;
Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 25 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*Eulerian(n+1, k) - 1 };
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Apr 25 2019
(Magma) Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
[[2*Eulerian(n+1, k)-1: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Apr 25 2019
(SageMath)
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*Eulerian(n+1, k)-1
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Apr 25 2019

Crossrefs

Cf. A008292, A109128.

Sequence in context: A174033 A119727 A157272 * A046739 A056752 A053714

Adjacent sequences: A176197 A176198 A176199 * A176201 A176202 A176203

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 11 2010

Extensions

Edited by G. C. Greubel, Apr 25 2019

Status

approved