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
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
Programs
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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
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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
Comments