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

1,2

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.

Links

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

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.

Example

[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)

Mathematica

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

Prog

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

Crossrefs

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

Cf. A075501, A075503.

Sequence in context: A038267 A027466 A218017 * A052104 A144450 A051339

Adjacent sequences: A075499 A075500 A075501 * A075503 A075504 A075505

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Oct 02 2002

Status

approved