A157637 Triangle, T(n, k, m) = 1 if (k=0 or k=n), otherwise (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*A157636(n, k)*T(n-2, k-1, m) for m = 1, read by rows.

1, 1, 1, 1, 5, 1, 1, 16, 16, 1, 1, 42, 136, 42, 1, 1, 99, 816, 816, 99, 1, 1, 219, 3951, 10200, 3951, 219, 1, 1, 466, 16632, 94827, 94827, 16632, 466, 1, 1, 968, 63670, 716160, 1601070, 716160, 63670, 968, 1, 1, 1981, 228112, 4657522, 20836740, 20836740, 4657522, 228112, 1981, 1

Offset

0,5

Comments

For the case of m = 0 the triangle becomes T(n, k, 0) = A007318(n, k). - G. C. Greubel, Dec 13 2021

Links

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

Formula

T(n, k, m) = 1 if (k=0 or k=n), otherwise (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*A157636(n, k)*T(n-2, k-1, m) for m = 1.

T(n, k) = T(n, n-k). - G. C. Greubel, Dec 13 2021

Example

Triangle begins as:
  1;
  1,    1;
  1,    5,      1;
  1,   16,     16,       1;
  1,   42,    136,      42,        1;
  1,   99,    816,     816,       99,        1;
  1,  219,   3951,   10200,     3951,      219,       1;
  1,  466,  16632,   94827,    94827,    16632,     466,      1;
  1,  968,  63670,  716160,  1601070,   716160,   63670,    968,    1;
  1, 1981, 228112, 4657522, 20836740, 20836740, 4657522, 228112, 1981, 1;

Mathematica

A157636[n_, k_]:= If[k==0||k==n, 1, n*k*(n-k)/2];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k) +1)*T[n-1, k-1, m] + (m*k + 1)*T[n-1, k, m] + m*A157636[n, k]*T[n-2, k-1, m]];
Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Sage)
@CachedFunction
def A157636(n, k): return 1 if (k==0 or k==n) else n*k*(n-k)/2
def T(n, k, q): return 1 if (k==0 or k==n) else (q*(n-k) +1)*T(n-1, k-1, q) + (q*k + 1)*T(n-1, k, q) + q*A157636(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, Dec 13 2021

Crossrefs

Cf. A007318, A157523, A157636.

Sequence in context: A174044 A174159 A074060 * A157181 A347974 A029847

Adjacent sequences: A157634 A157635 A157636 * A157638 A157639 A157640

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 03 2009

Extensions

Edited by G. C. Greubel, Dec 13 2021

Status

approved