A142462 Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 7.
1, 1, 1, 1, 16, 1, 1, 143, 143, 1, 1, 1166, 4290, 1166, 1, 1, 9357, 90002, 90002, 9357, 1, 1, 74892, 1621383, 3960088, 1621383, 74892, 1, 1, 599179, 27016857, 134142043, 134142043, 27016857, 599179, 1, 1, 4793482, 431017552, 3923731798, 7780238494, 3923731798, 431017552, 4793482, 1
Offset: 1
Examples
Triangle begins as: 1; 1, 1; 1, 16, 1; 1, 143, 143, 1; 1, 1166, 4290, 1166, 1; 1, 9357, 90002, 90002, 9357, 1; 1, 74892, 1621383, 3960088, 1621383, 74892, 1; 1, 599179, 27016857, 134142043, 134142043, 27016857, 599179, 1;
Links
- G. C. Greubel, Rows n = 1..50 of the triangle, flattened
- G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_7(n,k).
Crossrefs
Programs
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Magma
function T(n,k,m) if k eq 1 or k eq n then return 1; else return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m); end if; return T; end function; A142462:= func< n,k | T(n,k,7) >; [A142462(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 17 2022
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Mathematica
T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k -m+1)*T[n-1, k, m]]; A142462[n_, k_]:= T[n,k,7]; Table[A142462[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 17 2022 *)
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Sage
@CachedFunction def T(n,k,m): if (k==1 or k==n): return 1 else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m) def A142462(n,k): return T(n,k,7) flatten([[ A142462(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 17 2022
Formula
T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = 7.
Sum_{k=1..n} T(n, k) = A084947(n-1).
Extensions
Edited by N. J. A. Sloane, May 08 2013
Comments