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A075504 Stirling2 triangle with scaled diagonals (powers of 9).

%I A075504 #17 Mar 25 2017 20:47:30
%S A075504 1,9,1,81,27,1,729,567,54,1,6561,10935,2025,90,1,59049,203391,65610,
%T A075504 5265,135,1,531441,3720087,1974861,255150,11340,189,1,4782969,
%U A075504 67493007,57041334,11160261,765450,21546,252,1
%N A075504 Stirling2 triangle with scaled diagonals (powers of 9).
%C A075504 This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
%C A075504 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 A075504 Row sums give A075508(n), n >= 1. The columns (without leading zeros) give A001019 (powers of 9), A076008-A076013 for m=1..7.
%H A075504 Andrew Howroyd, <a href="/A075504/b075504.txt">Table of n, a(n) for n = 1..1275</a>
%F A075504 a(n, m) = (9^(n-m)) * stirling2(n, m).
%F A075504 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 A075504 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 A075504 G.f. for m-th column: (x^m)/Product_{k=1..m}(1-9k*x), m >= 1.
%F A075504 E.g.f. for m-th column: (((exp(9x) - 1)/9)^m)/m!, m >= 1.
%e A075504 [1]; [9,1]; [81,27,1]; ...; p(3,x) = x(81 + 27*x + x^2).
%e A075504 From _Andrew Howroyd_, Mar 25 2017: (Start)
%e A075504 Triangle starts
%e A075504 *       1
%e A075504 *       9        1
%e A075504 *      81       27        1
%e A075504 *     729      567       54        1
%e A075504 *    6561    10935     2025       90      1
%e A075504 *   59049   203391    65610     5265    135     1
%e A075504 *  531441  3720087  1974861   255150  11340   189   1
%e A075504 * 4782969 67493007 57041334 11160261 765450 21546 252 1
%e A075504 (End)
%t A075504 Flatten[Table[9^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* _Indranil Ghosh_, Mar 25 2017 *)
%o A075504 (PARI) for(n=1, 11, for(m=1, n, print1(9^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017
%Y A075504 Cf. A075503, A075505.
%Y A075504 Columns 2-7 are A076008-A076013.
%K A075504 nonn,easy,tabl
%O A075504 1,2
%A A075504 _Wolfdieter Lang_, Oct 02 2002