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%I #20 Mar 15 2022 03:03:38
%S 1,1,1,1,12,1,1,83,83,1,1,514,1826,514,1,1,3105,28310,28310,3105,1,1,
%T 18656,376615,905920,376615,18656,1,1,111967,4627821,22403635,
%U 22403635,4627821,111967,1,1,671838,54377008,478781506,940952670,478781506,54377008,671838,1
%N 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 = 5.
%C One of a family of triangles. For m = ...,-2,-1,0,1,2,3,4,5,... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142560, ...
%H G. C. Greubel, <a href="/A142460/b142460.txt">Rows n = 1..50 of the triangle, flattened</a>
%H 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_5(n,k).
%F 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(t,1,m) = T(n,n,m) = 1, and m = 5.
%F Sum_{k=1..n} T(n, k, 5) = A047055(n-1).
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 12, 1;
%e 1, 83, 83, 1;
%e 1, 514, 1826, 514, 1;
%e 1, 3105, 28310, 28310, 3105, 1;
%e 1, 18656, 376615, 905920, 376615, 18656, 1;
%e 1, 111967, 4627821, 22403635, 22403635, 4627821, 111967, 1;
%e 1, 671838, 54377008, 478781506, 940952670, 478781506, 54377008, 671838, 1;
%p A142460 := proc(n, k) if n = k then 1; elif k > n or k < 1 then 0 ; else (5*n-5*k+1)*procname(n-1, k-1)+(5*k-4)*procname(n-1, k) ; end if; end proc:
%p seq(seq(A142459(n, k), k=1..n), n=1..10) ; # _R. J. Mathar_, May 11 2013
%t 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 Table[T[n, k, 5], {n, 1, 10}, {k, 1, n}]//Flatten (* modified by _G. C. Greubel_, Mar 14 2022 *)
%o (Sage)
%o def T(n,k,m): # A142460
%o if (k==1 or k==n): return 1
%o else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
%o flatten([[T(n,k,5) for k in (1..n)] for n in (1..10)]) # _G. C. Greubel_, Mar 14 2022
%Y Cf. A225372 (m=-2), A144431 (m=-1), A007318 (m=0), A008292 (m=1), A060187 (m=2), A142458 (m=3), A142459 (m=4), this sequence (m=5), A142561 (m=6), A142562 (m=7), A167884 (m=8), A257608 (m=9).
%Y Cf. A047055 (row sums).
%K nonn,tabl,easy
%O 1,5
%A _Roger L. Bagula_, Sep 19 2008
%E Edited by _N. J. A. Sloane_, May 08 2013, May 11 2013
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