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

0,3

Links

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

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)

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A008292, A060187, A159041.

Sequence in context: A157629 A154336 A174109 * A152971 A142459 A157641

Adjacent sequences: A171689 A171690 A171691 * A171693 A171694 A171695

Keyword

nonn,tabf

Author

Roger L. Bagula, Dec 15 2009

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

Status

approved