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
Examples
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;
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
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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
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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 *) -
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 -
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*Eulerian(n+1,k)-1 [[T(n,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Apr 25 2019
Formula
T(n, m) = 2*Eulerian(n+1, m) - 1, where Eulerian(n, k) = A008292(n,k).
Extensions
Edited by G. C. Greubel, Apr 25 2019
Comments