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A075497 Stirling2 triangle with scaled diagonals (powers of 2).

%I A075497 #92 Feb 24 2025 08:52:17
%S A075497 1,2,1,4,6,1,8,28,12,1,16,120,100,20,1,32,496,720,260,30,1,64,2016,
%T A075497 4816,2800,560,42,1,128,8128,30912,27216,8400,1064,56,1,256,32640,
%U A075497 193600,248640,111216,21168,1848,72,1
%N A075497 Stirling2 triangle with scaled diagonals (powers of 2).
%C A075497 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 A075497 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(2*z) - 1)*x/2) - 1.
%C A075497 Subtriangle of (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, 12, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 13 2013
%C A075497 Also the inverse Bell transform of the double factorial of even numbers Product_ {k=0..n-1} (2*k+2) (A000165). For the definition of the Bell transform see A264428 and for cross-references A265604. - _Peter Luschny_, Dec 31 2015
%C A075497 This is the exponential Riordan array [exp(2*x), (exp(2*x) - 1)/2] belonging to the derivative subgroup of the exponential Riordan group. In the notation of Corcino, this is the triangle of (2, 2)-Stirling numbers of the second kind. A factorization of the array as an infinite product is given in the example section. - _Peter Bala_, Feb 20 2025
%H A075497 Alois P. Heinz, <a href="/A075497/b075497.txt">Rows n = 1..141, flattened</a>
%H A075497 Peter Bala, <a href="/A048993/a048993.pdf">The white diamond product of power series</a>
%H A075497 Peter Bala, <a href="/A143494/a143494.pdf">Factorising (r,b)-Stirling arrays</a>
%H A075497 Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H A075497 John R. Britnell and Mark Wildon, <a href="http://arxiv.org/abs/1507.04803">Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D</a>, arXiv 1507.04803 [math.CO], 2015.
%H A075497 Roberto B. Corcino, <a href="https://www.researchgate.net/publication/354943509">The (r, β)-Stirling Numbers</a>, The Mindanao Forum, Vol. XIV, No.2, pp. 91-99, 1999.
%H A075497 Roberto B. Corcino and Maribeth B. Montero, <a href="http://mathsociety.ph/matimyas/images/vol32/1/CorcinoMatimyas.pdf">The (r, β)-Stirling Numbers in the Context of 0-1 Tableau</a>, Jour. Math. Soc. of the Philippines, ISSN 0115-6926, Vol. 32, No. 1 (2009), pp. 45-52
%H A075497 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.
%H A075497 Wolfdieter Lang, <a href="/A075497/a075497.txt">First 10 rows</a>.
%H A075497 Toufik Mansour, <a href="https://arxiv.org/abs/math/0301157">Generalization of some identities involving the Fibonacci numbers</a>, arXiv:math/0301157 [math.CO], 2003.
%H A075497 Emanuele Munarini, <a href="https://doi.org/10.2298/AADM0901157M">Characteristic, admittance and matching polynomials of an antiregular graph</a>, Appl. Anal. Discrete Math 3 (1) (2009) 157-176.
%F A075497 a(n, m) = (2^(n-m)) * Stirling2(n, m).
%F A075497 a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*2)^(n-m))/(m-1)! for n >= m >= 1, else 0.
%F A075497 a(n, m) = 2*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 A075497 G.f. for m-th column: (x^m)/Product_{k=1..m}(1-2*k*x), m >= 1.
%F A075497 E.g.f. for m-th column: (((exp(2*x)-1)/2)^m)/m!, m >= 1.
%F A075497 The row polynomials in t are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+2*x)*d/dx. Cf. A008277. - _Peter Bala_, Nov 25 2011
%F A075497 From _Peter Bala_, Jan 13 2018: (Start)
%F A075497 n-th row polynomial R(n,x)= x o x o ... o x (n factors), where o is the deformed Hadamard product of power series defined in Bala, section 3.1.
%F A075497 R(n+1,x)/x = (x + 2) o (x + 2) o...o (x + 2) (n factors).
%F A075497 R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*2^(n-k)*R(k,x).
%F A075497 Dobinski-type formulas: R(n,x) = exp(-x/2)*Sum_{i >= 0} (2*i)^n* (x/2)^i/i!; 1/x*R(n+1,x) = exp(-x/2)*Sum_{i >= 0} (2 + 2*i)^n* (x/2)^i/i!. (End)
%e A075497 Triangle begins:
%e A075497   [1];
%e A075497   [2,1];
%e A075497   [4,6,1]; p(3,x) = x*(4 + 6*x + x^2).
%e A075497   ...;
%e A075497 Triangle (0, 2, 0, 4, 0, 6, 0, 8, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
%e A075497   1
%e A075497   0,  1
%e A075497   0,  2,   1
%e A075497   0,  4,   6,   1
%e A075497   0,  8,  28,  12,  1
%e A075497   0, 16, 120, 100, 20, 1. - _Philippe Deléham_, Feb 13 2013
%e A075497 From _Peter Bala_, Feb 23 2025: (Start)
%e A075497 The array factorizes as
%e A075497 / 1               \       /1             \ /1             \ /1            \
%e A075497 | 2    1           |     | 2   1          ||0  1           ||0  1          |
%e A075497 | 4    6   1       |  =  | 4   4   1      ||0  2   1       ||0  0  1       | ...
%e A075497 | 8   28  12   1   |     | 8  12   6  1   ||0  4   4  1    ||0  0  2  1    |
%e A075497 |16  120 100  20  1|     |16  32  24  8  1||0  8  12  6  1 ||0  0  4  4  1 |
%e A075497 |...               |     |...             ||...            ||...           |
%e A075497 where, in the infinite product on the right-hand side, the first array is the Riordan array (1/(1 - 2*x), x/(1 - 2*x)) = P^2, where P denotes Pascal's triangle. See A038207. Cf. A143494. (End)
%p A075497 with(combinat):
%p A075497 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A075497        `if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(
%p A075497         binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))
%p A075497     end:
%p A075497 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
%p A075497 seq(T(n), n=1..12);  # _Alois P. Heinz_, Aug 13 2015
%p A075497 # Alternatively, giving the triangle in the form displayed in the Example section:
%p A075497 gf := exp(x*exp(z)*sinh(z)):
%p A075497 X := n -> series(gf, z, n+2):
%p A075497 Z := n -> n!*expand(simplify(coeff(X(n), z, n))):
%p A075497 A075497_row := n -> op(PolynomialTools:-CoefficientList(Z(n), x)):
%p A075497 seq(A075497_row(n), n=0..9); # _Peter Luschny_, Jan 14 2018
%t A075497 Table[(2^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* _Michael De Vlieger_, Dec 31 2015 *)
%o A075497 (Sage) # uses[inverse_bell_transform from A265605]
%o A075497 multifact_2_2 = lambda n: prod(2*k + 2 for k in (0..n-1))
%o A075497 inverse_bell_matrix(multifact_2_2, 9) # _Peter Luschny_, Dec 31 2015
%o A075497 (PARI)
%o A075497 for(n=1, 11, for(m=1, n, print1(2^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017
%Y A075497 Columns 1-7 are A000079, A006516, A016283, A025966, A075510-A075512.
%Y A075497 Row sums are A004211.
%Y A075497 Cf. A008277, A075498-A075505.
%K A075497 nonn,easy,tabl
%O A075497 1,2
%A A075497 _Wolfdieter Lang_, Oct 02 2002