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A075502 Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7). 10

%I

%S 1,7,1,49,21,1,343,343,42,1,2401,5145,1225,70,1,16807,74431,30870,

%T 3185,105,1,117649,1058841,722701,120050,6860,147,1,823543,14941423,

%U 16235562,4084101,360150,13034,196,1

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

%C 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 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 Andrew Howroyd, <a href="/A075502/b075502.txt">Table of n, a(n) for n = 1..1275</a>

%F a(n, m) = (7^(n-m)) * stirling2(n, m).

%F 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 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 G.f. for m-th column: (x^m)/Product_{k=1..m}(1-7*k*x), m >= 1.

%F E.g.f. for m-th column: (((exp(7*x)-1)/7)^m)/m!, m >= 1.

%e [1]; [7,1]; [49,21,1]; ...; p(3,x) = x * (49 + 21*x + x^2).

%e From _Andrew Howroyd_, Mar 25 2017: (Start)

%e Triangle starts

%e * 1

%e * 7 1

%e * 49 21 1

%e * 343 343 42 1

%e * 2401 5145 1225 70 1

%e * 16807 74431 30870 3185 105 1

%e * 117649 1058841 722701 120050 6860 147 1

%e * 823543 14941423 16235562 4084101 360150 13034 196 1

%e (End)

%t Flatten[Table[7^(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(7^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017

%Y Columns 1-7 are A000420, A075921-A075925, A076002. Row sums are A075506.

%Y Cf. A075501, A075503.

%K nonn,easy,tabl

%O 1,2

%A _Wolfdieter Lang_, Oct 02 2002

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Last modified September 18 22:48 EDT 2020. Contains 337174 sequences. (Running on oeis4.)