A171692 Triangle read by rows: absolute values of odd-numbered rows of A159041.

1, 1, 10, 1, 1, 56, 246, 56, 1, 1, 246, 4047, 11572, 4047, 246, 1, 1, 1012, 46828, 408364, 901990, 408364, 46828, 1012, 1, 1, 4082, 474189, 9713496, 56604978, 105907308, 56604978, 9713496, 474189, 4082, 1, 1, 16368, 4520946, 193889840, 2377852335, 10465410528, 17505765564, 10465410528, 2377852335, 193889840, 4520946, 16368, 1

Offset: 0

Author

Roger L. Bagula, Dec 15 2009

Keywords

• nonn
• tabf

Examples

			Irregular triangle begins as:
  1;
  1,   10,     1;
  1,   56,   246,     56,      1;
  1,  246,  4047,  11572,   4047,    246,     1;
  1, 1012, 46828, 408364, 901990, 408364, 46828, 1012, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Crossrefs

Cf. A008292, A060187, A159041.

Programs

Mathematica

(* First program *)
f[x_, y_, m_]:= 2^(m+1)*Exp[2^m*x]/((1 -y*Exp[x])*(1 +(2^(m+1) -1)*Exp[2^m*x]));
Table[CoefficientList[SeriesCoefficient[Series[((1-y)^(n+1)/(2*y))*n!*f[x, y, 0], {x,0,30}], n], y], {n, 2, 20, 2}]//Flatten (* modified by G. C. Greubel, Mar 18 2022 *)
(* Second program *)
A008292[n_, k_]:= Sum[(-1)^j*(k-j)^n*Binomial[n+1, j], {j,0,k}];
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], T[n, k-1] + (-1)^k*A008292[n+2, k+1], T[n, n-k] ]]; (* T = A159041 *)
A171692[n_, k_]:= Abs[T[2*n, k]];
Table[A171692[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, Mar 18 2022 *)

Sage

def A008292(n,k): return sum( (-1)^j*(k-j)^n*binomial(n+1,j) for j in (0..k) )
@CachedFunction
def A159041(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return A159041(n,k-1) + (-1)^k*A008292(n+2,k+1)
    else: return A159041(n,n-k)
def A171692(n,k): return abs( A159041(2*n, k) )
flatten([[A171692(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022

Formula

T(n, k) = coefficients of (g(x, y)), where g(x, y) = n! * ((1-y)^(n+1)/(2*y)) * f(x, y, 0), with f(x, y, m) = 2^(m+1)*exp(2^m*x)/((1 -y*exp(x))*(1 +(2^(m+1) -1)*exp(2^m*x))).

From G. C. Greubel, Mar 18 2022: (Start)

T(n, k) = abs( A159041(2*n, k) ).

T(n, n-k) = T(n, k). (End)

Extensions

Edited by N. J. A. Sloane, May 10 2013

More terms from Jean-François Alcover, Feb 14 2014

Edited by G. C. Greubel, Mar 18 2022