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A075502 Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).

%I A075502 #12 Mar 29 2017 02:59:16
%S A075502 1,7,1,49,21,1,343,343,42,1,2401,5145,1225,70,1,16807,74431,30870,
%T A075502 3185,105,1,117649,1058841,722701,120050,6860,147,1,823543,14941423,
%U A075502 16235562,4084101,360150,13034,196,1
%N A075502 Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).
%C A075502 This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
%C A075502 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(7*z) - 1)*x/7) - 1.
%H A075502 Andrew Howroyd, <a href="/A075502/b075502.txt">Table of n, a(n) for n = 1..1275</a>
%F A075502 a(n, m) = (7^(n-m)) * stirling2(n, m).
%F A075502 a(n, m) = 7*m*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 A075502 a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*7)^(n-m))/(m-1)! for n >= m >= 1, else 0.
%F A075502 G.f. for m-th column: (x^m)/Product_{k=1..m}(1-7*k*x), m >= 1.
%F A075502 E.g.f. for m-th column: (((exp(7*x)-1)/7)^m)/m!, m >= 1.
%e A075502 [1]; [7,1]; [49,21,1]; ...; p(3,x) = x * (49 + 21*x + x^2).
%e A075502 From _Andrew Howroyd_, Mar 25 2017: (Start)
%e A075502 Triangle starts
%e A075502 *      1
%e A075502 *      7        1
%e A075502 *     49       21        1
%e A075502 *    343      343       42       1
%e A075502 *   2401     5145     1225      70      1
%e A075502 *  16807    74431    30870    3185    105     1
%e A075502 * 117649  1058841   722701  120050   6860   147   1
%e A075502 * 823543 14941423 16235562 4084101 360150 13034 196 1
%e A075502 (End)
%t A075502 Flatten[Table[7^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* _Indranil Ghosh_, Mar 25 2017 *)
%o A075502 (PARI) for(n=1, 11, for(m=1, n, print1(7^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017
%Y A075502 Columns 1-7 are A000420, A075921-A075925, A076002. Row sums are A075506.
%Y A075502 Cf. A075501, A075503.
%K A075502 nonn,easy,tabl
%O A075502 1,2
%A A075502 _Wolfdieter Lang_, Oct 02 2002