A157523 Triangle T(n, k, q) = (q*(n-k) +1)*T(n-1, k-1, q) + (q*k+1)*T(n-1, k, q) + q*A157522(n, k)*T(n-2, k-1, q), with T(n, 0, q) = T(n, n, q) = 1 and q = 1, read by rows.

1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 37, 95, 37, 1, 1, 82, 463, 463, 82, 1, 1, 173, 1910, 3799, 1910, 173, 1, 1, 356, 7096, 25672, 25672, 7096, 356, 1, 1, 723, 24645, 150994, 260519, 150994, 24645, 723, 1, 1, 1458, 81499, 804875, 2259903, 2259903, 804875, 81499, 1458, 1

Offset: 0

Author

Roger L. Bagula, Mar 02 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    5,     1;
  1,   15,    15,      1;
  1,   37,    95,     37,       1;
  1,   82,   463,    463,      82,       1;
  1,  173,  1910,   3799,    1910,     173,      1;
  1,  356,  7096,  25672,   25672,    7096,    356,     1;
  1,  723, 24645, 150994,  260519,  150994,  24645,   723,    1;
  1, 1458, 81499, 804875, 2259903, 2259903, 804875, 81499, 1458, 1;
		

Links

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

Crossrefs

Cf. A007318 (q=0), this sequence (q=1).

Cf. A157522.

Programs

Mathematica

f[n_, k_]= 1 + If[k<=Floor[n/4], k, If[Floor[n/4]A157522[n_, k_]:= f[n,k] +f[n,n-k] -1;
T[n_, k_, q_]:= T[n,k,q]= If[k==0 || k==n, 1, (q*(n-k) +1)*T[n-1,k-1,q] + (q*k+1)*T[n-1, k, q] + q*A157522[n, k]*T[n-2,k-1,q]];
Table[T[n,k,1], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 23 2022 *)

Sage

def f(n, k):
    if (k <= (n//4)): return k+1
    elif ((n//4) < k <= (n//2)): return (n//2)-k+1
    elif ((n//2) < k <= (3*n//4)): return k+1-(n//2)
    else: return n-k+1
def A157522(n,k): return f(n,k) + f(n,n-k) - 1
@CachedFunction
def T(n, k, q):
    if (k==0 or k==n): return 1
    else: return (q*(n-k) +1)*T(n-1, k-1, q) + (q*k+1)*T(n-1, k, q) + q*A157522(n, k)*T(n-2, k-1, q);
flatten([[T(n,k,1) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jan 23 2022

Formula

T(n, k, q) = (q*(n-k) +1)*T(n-1, k-1, q) + (q*k+1)*T(n-1, k, q) + q*A157522(n, k)*T(n-2, k-1, q), with T(n, 0, q) = T(n, n, q) = 1 and q = 1.

Extensions

Edited by G. C. Greubel, Jan 23 2022