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A171692 Triangle read by rows: absolute values of odd-numbered rows of A159041. 9
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 29 01:41 EST 2023. Contains 359905 sequences. (Running on oeis4.)