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A075499 Stirling2 triangle with scaled diagonals (powers of 4).

%I A075499 #24 Mar 26 2020 18:00:08
%S A075499 1,4,1,16,12,1,64,112,24,1,256,960,400,40,1,1024,7936,5760,1040,60,1,
%T A075499 4096,64512,77056,22400,2240,84,1,16384,520192,989184,435456,67200,
%U A075499 4256,112,1,65536,4177920,12390400,7956480,1779456,169344,7392,144,1
%N A075499 Stirling2 triangle with scaled diagonals (powers of 4).
%C A075499 This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
%C A075499 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(4*z) - 1)*x/4) - 1
%C A075499 Also the inverse Bell transform of the quadruple factorial numbers 4^n*n! (A047053) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - _Peter Luschny_, Dec 31 2015
%H A075499 Andrew Howroyd, <a href="/A075499/b075499.txt">Table of n, a(n) for n = 1..1275</a>
%F A075499 a(n, m) = (4^(n-m)) * stirling2(n, m).
%F A075499 a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*4)^(n-m))/(m-1)! for n >= m >= 1, else 0.
%F A075499 a(n, m) = 4m*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 A075499 G.f. for m-th column: (x^m)/Product_{k=1..m}(1-4k*x), m >= 1.
%F A075499 E.g.f. for m-th column: (((exp(4x)-1)/4)^m)/m!, m >= 1.
%e A075499 [1]; [4,1]; [16,12,1]; ...; p(3,x) = x(16 + 12*x + x^2).
%e A075499 From _Andrew Howroyd_, Mar 25 2017: (Start)
%e A075499 Triangle starts
%e A075499 *     1
%e A075499 *     4      1
%e A075499 *    16     12      1
%e A075499 *    64    112     24      1
%e A075499 *   256    960    400     40     1
%e A075499 *  1024   7936   5760   1040    60    1
%e A075499 *  4096  64512  77056  22400  2240   84   1
%e A075499 * 16384 520192 989184 435456 67200 4256 112 1
%e A075499 (End)
%t A075499 Table[(4^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* _Michael De Vlieger_, Dec 31 2015 *)
%o A075499 (Sage) # uses[inverse_bell_transform from A265605]
%o A075499 # Adds a column 1,0,0,... at the left side of the triangle.
%o A075499 multifact_4_4 = lambda n: prod(4*k + 4 for k in (0..n-1))
%o A075499 inverse_bell_matrix(multifact_4_4, 9) # _Peter Luschny_, Dec 31 2015
%o A075499 (PARI)
%o A075499 for(n=1, 11, for(m=1, n, print1(4^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017
%Y A075499 Columns 1-7 are A000302, A016152, A019677, A075907-A075910. Row sums are A004213.
%Y A075499 Cf. A075498, A075500.
%K A075499 nonn,easy,tabl
%O A075499 1,2
%A A075499 _Wolfdieter Lang_, Oct 02 2002