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

Author

Roger L. Bagula, Mar 03 2009

Keywords

• nonn
• tabl

Comments

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

Examples

			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;
		

Links

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

Crossrefs

Cf. A007318, A157523, A157636.

Programs

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

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

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

Extensions

Edited by G. C. Greubel, Dec 13 2021