A225532 Triangle T(n, k) = abs(A225483(n/2, k)) if (n mod 2 = 0), otherwise abs(A225482((n-1)/2, k) - A225483((n-1)/2, k-1)), read by rows.

1, 1, 1, 1, 26, 1, 1, 27, 27, 1, 1, 120, 1192, 120, 1, 1, 121, 1312, 1312, 121, 1, 1, 502, 14609, 88736, 14609, 502, 1, 1, 503, 15111, 103345, 103345, 15111, 503, 1, 1, 2036, 152638, 2205524, 9890752, 2205524, 152638, 2036, 1, 1, 2037, 154674, 2358162, 12096276, 12096276, 2358162, 154674, 2037, 1

Offset

0,5

Links

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

Formula

From G. C. Greubel, Mar 29 2022: (Start)

T(n, k) = abs(A225483(n/2, k)) if (n mod 2 = 0), otherwise abs(A225482((n-1)/2, k) - A225483((n-1)/2, k-1)).

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

Example

Triangle begins:
  1;
  1,   1;
  1,  26,     1;
  1,  27,    27,      1;
  1, 120,  1192,    120,      1;
  1, 121,  1312,   1312,    121,     1;
  1, 502, 14609,  88736,  14609,   502,   1;
  1, 503, 15111, 103345, 103345, 15111, 503, 1;

Mathematica

(* First program *)
Needs["Combinatorica`"];
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);
q[n_, x_]= If[Mod[n, 2]==0, (1-x)*p[n/2, x], p[(n+1)/2, x]];
Table[Abs[CoefficientList[q[(4*n +(-1)^n +5)/2, x], x]], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 29 2022 *)
(* Second program *)
A008292[n_, k_]:= A008292[n, k]= Sum[(-1)^j*(k-j)^n*Binomial[n+1, j], {j, 0, k}];
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 *)
A225483[n_, k_]:= Sum[(-1)^(k-j)*f[2*n+1, j], {j, 0, k}];
T[n_, k_]:= If[Mod[n, 2]==0, A225483[n/2, k], A225483[(n-1)/2, k] - A225483[(n - 1)/2, k-1] ]//Abs;
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 29 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 f(n, k): # A159041
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return f(n, k-1) + (-1)^k*A008292(n+2, k+1)
    else: return f(n, n-k)
def A225483(n, k): return sum( (-1)^(k-j)*f(2*n+1, j) for j in (0..k) )
@CachedFunction
def A225532(n, k):
    if (n%2==0): return abs(A225483(n/2, k))
    else: return abs( A225483((n-1)/2, k) - A225483((n-1)/2, k-1) )
flatten([[A225532(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 29 2022

Crossrefs

Cf. A008292, A159041, A171692, A204621, A225483.

Sequence in context: A040678 A040677 A040676 * A177970 A225483 A183065

Adjacent sequences: A225529 A225530 A225531 * A225533 A225534 A225535

Keyword

nonn,tabl

Author

Roger L. Bagula, May 09 2013

Extensions

Edited by G. C. Greubel, Mar 29 2022

Status

approved