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

0,5

Links

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

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.

Example

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;

Mathematica

f[n_, k_]= 1 + If[k<=Floor[n/4], k, If[Floor[n/4]<k<=Floor[n/2], Floor[n/2]-k, If[Floor[n/2]<k<=Floor[3*n/4], k-Floor[n/2], n-k]]];
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 *)

Prog

(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

Crossrefs

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

Cf. A157522.

Sequence in context: A196019 A056940 A168288 * A141691 A157147 A347973

Adjacent sequences: A157520 A157521 A157522 * A157524 A157525 A157526

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 02 2009

Extensions

Edited by G. C. Greubel, Jan 23 2022

Status

approved