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A075500 Stirling2 triangle with scaled diagonals (powers of 5).

%I A075500 #19 Jun 23 2018 06:09:15
%S A075500 1,5,1,25,15,1,125,175,30,1,625,1875,625,50,1,3125,19375,11250,1625,
%T A075500 75,1,15625,196875,188125,43750,3500,105,1,78125,1984375,3018750,
%U A075500 1063125,131250,6650,140,1,390625,19921875
%N A075500 Stirling2 triangle with scaled diagonals (powers of 5).
%C A075500 This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
%C A075500 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(5*z) - 1)*x/5) - 1.
%H A075500 Andrew Howroyd, <a href="/A075500/b075500.txt">Table of n, a(n) for n = 1..1275</a>
%F A075500 a(n, m) = (5^(n-m)) * stirling2(n, m).
%F A075500 a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*5)^(n-m))/(m-1)! for n >= m >= 1, else 0.
%F A075500 a(n, m) = 5m*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 A075500 G.f. for m-th column: (x^m)/Product_{k=1..m}(1-5k*x), m >= 1.
%F A075500 E.g.f. for m-th column: (((exp(5x)-1)/5)^m)/m!, m >= 1.
%e A075500 [1]; [5,1]; [25,15,1]; ...; p(3,x) = x(25 + 15*x + x^2).
%e A075500 From _Andrew Howroyd_, Mar 25 2017: (Start)
%e A075500 Triangle starts
%e A075500 *     1
%e A075500 *     5       1
%e A075500 *    25      15       1
%e A075500 *   125     175      30       1
%e A075500 *   625    1875     625      50      1
%e A075500 *  3125   19375   11250    1625     75    1
%e A075500 * 15625  196875  188125   43750   3500  105   1
%e A075500 * 78125 1984375 3018750 1063125 131250 6650 140 1
%e A075500 (End)
%p A075500 # The function BellMatrix is defined in A264428.
%p A075500 # Adds (1,0,0,0, ..) as column 0.
%p A075500 BellMatrix(n -> 5^n, 9); # _Peter Luschny_, Jan 28 2016
%t A075500 Flatten[Table[5^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* _Indranil Ghosh_, Mar 25 2017 *)
%t A075500 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
%t A075500 rows = 10;
%t A075500 M = BellMatrix[5^#&, rows];
%t A075500 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)
%o A075500 (PARI) for(n=1, 11, for(m=1, n, print1(5^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017
%Y A075500 Columns 1-7 are A000351, A016164, A075911-A075915. Row sums are A005011(n-1).
%Y A075500 Cf. A075499, A075501.
%K A075500 nonn,easy,tabl
%O A075500 1,2
%A A075500 _Wolfdieter Lang_, Oct 02 2002