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A166973 Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.

%I A166973 #17 Aug 13 2017 05:04:12
%S A166973 1,1,1,1,7,1,1,43,18,1,1,259,241,34,1,1,1555,2910,785,55,1,1,9331,
%T A166973 33565,15470,1940,81,1,1,55987,378546,281085,56210,4046,112,1,1,
%U A166973 335923,4219993,4875906,1461495,161406,7518,148,1,1,2015539,46755846,82234489
%N A166973 Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.
%C A166973 The recursion T(n, k) = (m*n - m*k + 1)*T(n-1, k-1) + (5*k - 4)*(m*k - (m - 1))*T(n-1, k) was intended to range over m values 0 to 4 as given by the original Mathematica code. This sequences is the case for m = 0. - _G. C. Greubel_, May 29 2016
%C A166973 With offset 0 in the rows and columns this is the Sheffer triangle S2[5,1] = (exp(x), (exp(5*x) - 1)/5). See S2[4,1] = A111578 (with offsets 0), S[3,1] = A111577 (with offsets 0), S2[2,1] = A039755
%H A166973 G. C. Greubel, <a href="/A166973/b166973.txt">Table of n, a(n) for the first 25 rows</a>
%F A166973 T(n, k) = T(n - 1, k - 1) + (5*k - 4)*T(n - 1, k).
%F A166973 E.g.f. column k: int(exp(x)*((exp(5*x)-1)/5)^(k-1)/(k-1)!, x) + (-1)^k/A008548(k). - _Wolfdieter Lang_, Aug 13 2017
%e A166973 Triangle T(n, k) starts:
%e A166973 n\k   1       2        3        4        5       6      7     8   9 10 ...
%e A166973 1:    1
%e A166973 2:    1       1
%e A166973 3:    1       7        1
%e A166973 4:    1      43       18        1
%e A166973 5:    1     259      241       34        1
%e A166973 6:    1    1555     2910      785       55       1
%e A166973 7:    1    9331    33565    15470     1940      81      1
%e A166973 8:    1   55987   378546   281085    56210    4046    112     1
%e A166973 9:    1  335923  4219993  4875906  1461495  161406   7518   148   1
%e A166973 10:   1 2015539 46755846 82234489 35567301 5658051 394464 12846 189  1
%e A166973 ... Reformatted, - _Wolfdieter Lang_, Aug 13 2017
%t A166973 A[n_, 1] := 1; A[n_, n_] := 1; A[n_, k_] := A[n - 1, k - 1] + (5*k - 4)*A[n - 1,k]; Flatten[ Table[A[n, k], {n, 10}, {k, n}]] (* modified by _G. C. Greubel_, May 29 2016 *)
%Y A166973 Cf. A111577.
%Y A166973 S2[4,1] = A111578 (with offsets 0), S2[3,1] = A111577 (with offsets 0), S2[2,1] = A039755. - _Wolfdieter Lang_, Aug 13 2017
%K A166973 nonn,easy,tabl
%O A166973 1,5
%A A166973 _Roger L. Bagula_, Oct 26 2009