A075502 Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).

1, 7, 1, 49, 21, 1, 343, 343, 42, 1, 2401, 5145, 1225, 70, 1, 16807, 74431, 30870, 3185, 105, 1, 117649, 1058841, 722701, 120050, 6860, 147, 1, 823543, 14941423, 16235562, 4084101, 360150, 13034, 196, 1

Offset: 1

Author

Wolfdieter Lang, Oct 02 2002

Keywords

• nonn
• easy
• tabl

Comments

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(7*z) - 1)*x/7) - 1.

Examples

			[1]; [7,1]; [49,21,1]; ...; p(3,x) = x * (49 + 21*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*      1
*      7        1
*     49       21        1
*    343      343       42       1
*   2401     5145     1225      70      1
*  16807    74431    30870    3185    105     1
* 117649  1058841   722701  120050   6860   147   1
* 823543 14941423 16235562 4084101 360150 13034 196 1
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Columns 1-7 are A000420, A075921-A075925, A076002. Row sums are A075506.

Cf. A075501, A075503.

Programs

Mathematica

Flatten[Table[7^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

PARI

for(n=1, 11, for(m=1, n, print1(7^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n, m) = (7^(n-m)) * stirling2(n, m).

a(n, m) = 7*m*a(n-1, m) + a(n-1, m-1), n>=m>=1, else 0, with a(n, 0) := 0 and a(1, 1)=1.

a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*7)^(n-m))/(m-1)! for n >= m >= 1, else 0.

G.f. for m-th column: (x^m)/Product_{k=1..m}(1-7*k*x), m >= 1.

E.g.f. for m-th column: (((exp(7*x)-1)/7)^m)/m!, m >= 1.