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%I #16 Mar 14 2024 11:12:44
%S 1,10,1,100,30,1,1000,700,60,1,10000,15000,2500,100,1,100000,310000,
%T 90000,6500,150,1,1000000,6300000,3010000,350000,14000,210,1,10000000,
%U 127000000,96600000,17010000,1050000,26600,280,1
%N Stirling2 triangle with scaled diagonals (powers of 10).
%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(10*z) - 1)*x/10) - 1.
%H Andrew Howroyd, <a href="/A075505/b075505.txt">Table of n, a(n) for n = 1..1275</a>
%H 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 a(n, m) = (10^(n-m)) * stirling2(n, m).
%F 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 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 G.f. for m-th column: (x^m)/Product_{k=1..m} (1-10k*x), m >= 1.
%F E.g.f. for m-th column: (((exp(10x)-1)/10)^m)/m!, m >= 1.
%e [1]; [10,1]; [100,30,1]; ...; p(3,x) = x(100 + 30*x + x^2).
%e From _Andrew Howroyd_, Mar 25 2017: (Start)
%e Triangle starts
%e * 1
%e * 10 1
%e * 100 30 1
%e * 1000 700 60 1
%e * 10000 15000 2500 100 1
%e * 100000 310000 90000 6500 150 1
%e * 1000000 6300000 3010000 350000 14000 210 1
%e * 10000000 127000000 96600000 17010000 1050000 26600 280 1
%e (End)
%t Flatten[Table[10^(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(10^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017
%Y Row sums are A075509.
%Y Cf. A075504.
%K nonn,easy,tabl
%O 1,2
%A _Wolfdieter Lang_, Oct 02 2002
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