A225483 - OEIS

%I #17 Mar 31 2022 10:17:43

%S 1,1,-26,1,1,-120,1192,-120,1,1,-502,14609,-88736,14609,-502,1,1,

%T -2036,152638,-2205524,9890752,-2205524,152638,-2036,1,1,-8178,

%U 1479727,-45541628,424761262,-1551163136,424761262,-45541628,1479727,-8178,1

%N Triangle T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A159041(2*n+1, j), read by rows.

%H G. C. Greubel, <a href="/A225483/b225483.txt">Rows n = 0..50 if the irregular triangle, flattened</a>

%F T(n, k) = [x^k]( A159041(x,n)/(x+1) ).

%F From _G. C. Greubel_, Mar 29 2022: (Start)

%F T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A159041(2*n+1, j).

%F T(n, 2*n-k) = T(n, k). (End)

%e The triangle begins:

%e   1;

%e   1,   -26,      1;

%e   1,  -120,   1192,     -120,       1;

%e   1,  -502,  14609,   -88736,   14609,     -502,      1;

%e   1, -2036, 152638, -2205524, 9890752, -2205524, 152638, -2036, 1;

%t (* First program *)

%t Needs["Combinatorica`"];

%t p[n_, x_]:= p[n,x]= Sum[If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i<=Floor[n/2], (-1)^i*Eulerian[n+1, i]*x^i, (-1)^(n-i+1)*Eulerian[n+1, i]*x^i]], {i,0,n}]/(1- x^2);

%t Table[CoefficientList[p[x, 2*n], x], {n,0,10}]//Flatten

%t (* Second program *)

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

%t f[n_, k_]:= f[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], f[n, k-1] + (-1)^k*A008292[n+2, k+1], f[n, n-k]]]; (* f = A159041 *)

%t T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j)*f[2*n+1,j], {j,0,k}];

%t Table[T[n, k], {n,0,10}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Mar 29 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 f(n, k): # A159041

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

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

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

%o def A225483(n,k): return sum( (-1)^(k-j)*f(2*n+1,j) for j in (0..k) )

%o flatten([[A225483(n, k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Mar 29 2022

%Y Cf. A008292, A060187, A158041.

%K sign,tabf

%O 0,3

%A _Roger L. Bagula_, May 08 2013

%E Edited by _G. C. Greubel_, Mar 29 2022