A157149 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 3, read by rows.
1, 1, 1, 1, 11, 1, 1, 57, 57, 1, 1, 247, 930, 247, 1, 1, 1013, 10006, 10006, 1013, 1, 1, 4083, 89139, 225230, 89139, 4083, 1, 1, 16369, 719691, 3771323, 3771323, 719691, 16369, 1, 1, 65519, 5495836, 53239541, 108865438, 53239541, 5495836, 65519, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, 11, 1; 1, 57, 57, 1; 1, 247, 930, 247, 1; 1, 1013, 10006, 10006, 1013, 1; 1, 4083, 89139, 225230, 89139, 4083, 1; 1, 16369, 719691, 3771323, 3771323, 719691, 16369, 1; 1, 65519, 5495836, 53239541, 108865438, 53239541, 5495836, 65519, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
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Maple
A157149 := proc(n,k) option remember; if k < 0 or k> n then 0; elif k = 0 or k = n then 1; else (3*(n-k)+1)*procname(n-1,k-1) + (3*k+1)*procname(n-1,k) + 3*k*(n-k)*procname(n-2,k-1); end if; end proc: seq(seq(A157149(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Feb 06 2015
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Mathematica
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*k*(n-k)*T[n-2,k-1,m]]; Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *) -
Sage
@CachedFunction def T(n,k,m): # A157149 if (k==0 or k==n): return 1 else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*k*(n-k)*T(n-2,k-1,m) flatten([[T(n,k,3) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 09 2022
Formula
T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 3.
T(n, n-k, 3) = T(n, k, 3).
T(n, 1, 3) = A289255(n). - G. C. Greubel, Jan 09 2022
Extensions
Edited by G. C. Greubel, Jan 09 2022