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A075505 Stirling2 triangle with scaled diagonals (powers of 10).

%I A075505 #16 Mar 14 2024 11:12:44
%S A075505 1,10,1,100,30,1,1000,700,60,1,10000,15000,2500,100,1,100000,310000,
%T A075505 90000,6500,150,1,1000000,6300000,3010000,350000,14000,210,1,10000000,
%U A075505 127000000,96600000,17010000,1050000,26600,280,1
%N A075505 Stirling2 triangle with scaled diagonals (powers of 10).
%C A075505 This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
%C A075505 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(10*z) - 1)*x/10) - 1.
%H A075505 Andrew Howroyd, <a href="/A075505/b075505.txt">Table of n, a(n) for n = 1..1275</a>
%H A075505 Paweł Hitczenko, <a href="https://arxiv.org/abs/2403.03422">A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality</a>, arXiv:2403.03422 [math.CO], 2024. See p. 8.
%F A075505 a(n, m) = (10^(n-m)) * stirling2(n, m).
%F A075505 a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*10)^(n-m))/(m-1)! for n >= m >= 1, else 0.
%F A075505 a(n, m) = 10m*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 A075505 G.f. for m-th column: (x^m)/Product_{k=1..m} (1-10k*x), m >= 1.
%F A075505 E.g.f. for m-th column: (((exp(10x)-1)/10)^m)/m!, m >= 1.
%e A075505 [1]; [10,1]; [100,30,1]; ...; p(3,x) = x(100 + 30*x + x^2).
%e A075505 From _Andrew Howroyd_, Mar 25 2017: (Start)
%e A075505 Triangle starts
%e A075505 *        1
%e A075505 *       10         1
%e A075505 *      100        30        1
%e A075505 *     1000       700       60        1
%e A075505 *    10000     15000     2500      100       1
%e A075505 *   100000    310000    90000     6500     150     1
%e A075505 *  1000000   6300000  3010000   350000   14000   210   1
%e A075505 * 10000000 127000000 96600000 17010000 1050000 26600 280 1
%e A075505 (End)
%t A075505 Flatten[Table[10^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* _Indranil Ghosh_, Mar 25 2017 *)
%o A075505 (PARI) for(n=1, 11, for(m=1, n, print1(10^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017
%Y A075505 Row sums are A075509.
%Y A075505 Cf. A075504.
%K A075505 nonn,easy,tabl
%O A075505 1,2
%A A075505 _Wolfdieter Lang_, Oct 02 2002