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.

1, 1, 1, 1, 7, 1, 1, 43, 18, 1, 1, 259, 241, 34, 1, 1, 1555, 2910, 785, 55, 1, 1, 9331, 33565, 15470, 1940, 81, 1, 1, 55987, 378546, 281085, 56210, 4046, 112, 1, 1, 335923, 4219993, 4875906, 1461495, 161406, 7518, 148, 1, 1, 2015539, 46755846, 82234489

Offset: 1

Author

Roger L. Bagula, Oct 26 2009

Keywords

• nonn
• easy
• tabl

Comments

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

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

Examples

			Triangle T(n, k) starts:
n\k   1       2        3        4        5       6      7     8   9 10 ...
1:    1
2:    1       1
3:    1       7        1
4:    1      43       18        1
5:    1     259      241       34        1
6:    1    1555     2910      785       55       1
7:    1    9331    33565    15470     1940      81      1
8:    1   55987   378546   281085    56210    4046    112     1
9:    1  335923  4219993  4875906  1461495  161406   7518   148   1
10:   1 2015539 46755846 82234489 35567301 5658051 394464 12846 189  1
... Reformatted, - _Wolfdieter Lang_, Aug 13 2017
		

Links

• G. C. Greubel, Table of n, a(n) for the first 25 rows

Crossrefs

Cf. A111577.

S2[4,1] = A111578 (with offsets 0), S2[3,1] = A111577 (with offsets 0), S2[2,1] = A039755. - Wolfdieter Lang, Aug 13 2017

Programs

Mathematica

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 *)

Formula

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

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