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A142461 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 = 6.

%I A142461 #15 Mar 17 2022 23:46:25
%S A142461 1,1,1,1,14,1,1,111,111,1,1,796,2886,796,1,1,5597,52642,52642,5597,1,
%T A142461 1,39210,824271,2000396,824271,39210,1,1,274507,11931033,58614299,
%U A142461 58614299,11931033,274507,1,1,1921592,165260188,1483533704,2930714950,1483533704,165260188,1921592,1
%N A142461 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 = 6.
%H A142461 G. C. Greubel, <a href="/A142461/b142461.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A142461 G. Strasser, <a href="http://dx.doi.org/10.1017/S0305004110000538">Generalisation of the Euler adic</a>, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_6(n,k).
%F A142461 T(n,k,m) = (m*n - m*k + 1)*T(n-1, k-1, m) + (m*k - (m-1))*T(n-1, k, m), with T(n, 1, m) = T(n, n, m) = 1, and m = 6.
%F A142461 Sum_{k=1..n} T(n, k, 6) = A047657(n-1).
%e A142461 Triangle begins as:
%e A142461   1;
%e A142461   1,      1;
%e A142461   1,     14,        1;
%e A142461   1,    111,      111,        1;
%e A142461   1,    796,     2886,      796,        1;
%e A142461   1,   5597,    52642,    52642,     5597,        1;
%e A142461   1,  39210,   824271,  2000396,   824271,    39210,      1;
%e A142461   1, 274507, 11931033, 58614299, 58614299, 11931033, 274507, 1;
%t A142461 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]];
%t A142461 A142461[n_, k_]:= T[n, k, 6];
%t A142461 Table[A142461[n, k], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 17 2022 *)
%o A142461 (Sage)
%o A142461 @CachedFunction
%o A142461 def T(n,k,m):
%o A142461     if (k==1 or k==n): return 1
%o A142461     else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
%o A142461 def A142461(n,k): return T(n,k,6)
%o A142461 flatten([[ A142461(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 17 2022
%Y A142461 For m = ...,-2,-1,0,1,2,3,4,5,6,7, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142460, ...
%Y A142461 Cf. A047657 (row sums).
%K A142461 nonn,tabl,easy
%O A142461 1,5
%A A142461 _Roger L. Bagula_, Sep 19 2008
%E A142461 Edited by _N. J. A. Sloane_, May 08 2013