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Triangle read by rows: absolute values of odd-numbered rows of A159041.
9

%I #31 Mar 19 2022 02:32:41

%S 1,1,10,1,1,56,246,56,1,1,246,4047,11572,4047,246,1,1,1012,46828,

%T 408364,901990,408364,46828,1012,1,1,4082,474189,9713496,56604978,

%U 105907308,56604978,9713496,474189,4082,1,1,16368,4520946,193889840,2377852335,10465410528,17505765564,10465410528,2377852335,193889840,4520946,16368,1

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

%H G. C. Greubel, <a href="/A171692/b171692.txt">Rows n = 0..50 of the irregular triangle, flattened</a>

%F 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))).

%F From _G. C. Greubel_, Mar 18 2022: (Start)

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

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

%e Irregular triangle begins as:

%e 1;

%e 1, 10, 1;

%e 1, 56, 246, 56, 1;

%e 1, 246, 4047, 11572, 4047, 246, 1;

%e 1, 1012, 46828, 408364, 901990, 408364, 46828, 1012, 1;

%t (* First program *)

%t f[x_, y_, m_]:= 2^(m+1)*Exp[2^m*x]/((1 -y*Exp[x])*(1 +(2^(m+1) -1)*Exp[2^m*x]));

%t 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 *)

%t (* Second program *)

%t A008292[n_, k_]:= Sum[(-1)^j*(k-j)^n*Binomial[n+1, j], {j,0,k}];

%t 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 *)

%t A171692[n_, k_]:= Abs[T[2*n, k]];

%t Table[A171692[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Mar 18 2022 *)

%o (Sage)

%o def A008292(n,k): return sum( (-1)^j*(k-j)^n*binomial(n+1,j) for j in (0..k) )

%o @CachedFunction

%o def A159041(n,k):

%o if (k==0 or k==n): return 1

%o elif (k <= (n//2)): return A159041(n,k-1) + (-1)^k*A008292(n+2,k+1)

%o else: return A159041(n,n-k)

%o def A171692(n,k): return abs( A159041(2*n, k) )

%o flatten([[A171692(n,k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Mar 18 2022

%Y Cf. A008292, A060187, A159041.

%K nonn,tabf

%O 0,3

%A _Roger L. Bagula_, Dec 15 2009

%E Edited by _N. J. A. Sloane_, May 10 2013

%E More terms from _Jean-François Alcover_, Feb 14 2014

%E Edited by _G. C. Greubel_, Mar 18 2022