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Stirling2 triangle with scaled diagonals (powers of 9).
10

%I #17 Mar 25 2017 20:47:30

%S 1,9,1,81,27,1,729,567,54,1,6561,10935,2025,90,1,59049,203391,65610,

%T 5265,135,1,531441,3720087,1974861,255150,11340,189,1,4782969,

%U 67493007,57041334,11160261,765450,21546,252,1

%N Stirling2 triangle with scaled diagonals (powers of 9).

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

%C 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(9*z) - 1)*x/9) - 1.

%C Row sums give A075508(n), n >= 1. The columns (without leading zeros) give A001019 (powers of 9), A076008-A076013 for m=1..7.

%H Andrew Howroyd, <a href="/A075504/b075504.txt">Table of n, a(n) for n = 1..1275</a>

%F a(n, m) = (9^(n-m)) * stirling2(n, m).

%F a(n, m) = Sum_{p=0..m-1} (A075513(m, p)*((p+1)*9)^(n-m))/(m-1)! for n >= m >= 1, else 0.

%F a(n, m) = 9m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.

%F G.f. for m-th column: (x^m)/Product_{k=1..m}(1-9k*x), m >= 1.

%F E.g.f. for m-th column: (((exp(9x) - 1)/9)^m)/m!, m >= 1.

%e [1]; [9,1]; [81,27,1]; ...; p(3,x) = x(81 + 27*x + x^2).

%e From _Andrew Howroyd_, Mar 25 2017: (Start)

%e Triangle starts

%e * 1

%e * 9 1

%e * 81 27 1

%e * 729 567 54 1

%e * 6561 10935 2025 90 1

%e * 59049 203391 65610 5265 135 1

%e * 531441 3720087 1974861 255150 11340 189 1

%e * 4782969 67493007 57041334 11160261 765450 21546 252 1

%e (End)

%t Flatten[Table[9^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* _Indranil Ghosh_, Mar 25 2017 *)

%o (PARI) for(n=1, 11, for(m=1, n, print1(9^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017

%Y Cf. A075503, A075505.

%Y Columns 2-7 are A076008-A076013.

%K nonn,easy,tabl

%O 1,2

%A _Wolfdieter Lang_, Oct 02 2002