%I #17 Aug 13 2017 05:04:12

%S 1,1,1,1,7,1,1,43,18,1,1,259,241,34,1,1,1555,2910,785,55,1,1,9331,

%T 33565,15470,1940,81,1,1,55987,378546,281085,56210,4046,112,1,1,

%U 335923,4219993,4875906,1461495,161406,7518,148,1,1,2015539,46755846,82234489

%N 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 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 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 G. C. Greubel, <a href="/A166973/b166973.txt">Table of n, a(n) for the first 25 rows</a>

%F T(n, k) = T(n - 1, k - 1) + (5*k - 4)*T(n - 1, k).

%F 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 Triangle T(n, k) starts:

%e n\k 1 2 3 4 5 6 7 8 9 10 ...

%e 1: 1

%e 2: 1 1

%e 3: 1 7 1

%e 4: 1 43 18 1

%e 5: 1 259 241 34 1

%e 6: 1 1555 2910 785 55 1

%e 7: 1 9331 33565 15470 1940 81 1

%e 8: 1 55987 378546 281085 56210 4046 112 1

%e 9: 1 335923 4219993 4875906 1461495 161406 7518 148 1

%e 10: 1 2015539 46755846 82234489 35567301 5658051 394464 12846 189 1

%e ... Reformatted, - _Wolfdieter Lang_, Aug 13 2017

%t 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 Cf. A111577.

%Y S2[4,1] = A111578 (with offsets 0), S2[3,1] = A111577 (with offsets 0), S2[2,1] = A039755. - _Wolfdieter Lang_, Aug 13 2017

%K nonn,easy,tabl

%O 1,5

%A _Roger L. Bagula_, Oct 26 2009