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%I #92 Feb 24 2025 08:52:17
%S 1,2,1,4,6,1,8,28,12,1,16,120,100,20,1,32,496,720,260,30,1,64,2016,
%T 4816,2800,560,42,1,128,8128,30912,27216,8400,1064,56,1,256,32640,
%U 193600,248640,111216,21168,1848,72,1
%N Stirling2 triangle with scaled diagonals (powers of 2).
%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(2*z) - 1)*x/2) - 1.
%C 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 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 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 Alois P. Heinz, <a href="/A075497/b075497.txt">Rows n = 1..141, flattened</a>
%H Peter Bala, <a href="/A048993/a048993.pdf">The white diamond product of power series</a>
%H Peter Bala, <a href="/A143494/a143494.pdf">Factorising (r,b)-Stirling arrays</a>
%H 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 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 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 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 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 Wolfdieter Lang, <a href="/A075497/a075497.txt">First 10 rows</a>.
%H 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 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 a(n, m) = (2^(n-m)) * Stirling2(n, m).
%F 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 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 G.f. for m-th column: (x^m)/Product_{k=1..m}(1-2*k*x), m >= 1.
%F E.g.f. for m-th column: (((exp(2*x)-1)/2)^m)/m!, m >= 1.
%F 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 From _Peter Bala_, Jan 13 2018: (Start)
%F 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 R(n+1,x)/x = (x + 2) o (x + 2) o...o (x + 2) (n factors).
%F R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*2^(n-k)*R(k,x).
%F 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 Triangle begins:
%e [1];
%e [2,1];
%e [4,6,1]; p(3,x) = x*(4 + 6*x + x^2).
%e ...;
%e Triangle (0, 2, 0, 4, 0, 6, 0, 8, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
%e 1
%e 0, 1
%e 0, 2, 1
%e 0, 4, 6, 1
%e 0, 8, 28, 12, 1
%e 0, 16, 120, 100, 20, 1. - _Philippe Deléham_, Feb 13 2013
%e From _Peter Bala_, Feb 23 2025: (Start)
%e The array factorizes as
%e / 1 \ /1 \ /1 \ /1 \
%e | 2 1 | | 2 1 ||0 1 ||0 1 |
%e | 4 6 1 | = | 4 4 1 ||0 2 1 ||0 0 1 | ...
%e | 8 28 12 1 | | 8 12 6 1 ||0 4 4 1 ||0 0 2 1 |
%e |16 120 100 20 1| |16 32 24 8 1||0 8 12 6 1 ||0 0 4 4 1 |
%e |... | |... ||... ||... |
%e 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 with(combinat):
%p b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p `if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(
%p binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
%p seq(T(n), n=1..12); # _Alois P. Heinz_, Aug 13 2015
%p # Alternatively, giving the triangle in the form displayed in the Example section:
%p gf := exp(x*exp(z)*sinh(z)):
%p X := n -> series(gf, z, n+2):
%p Z := n -> n!*expand(simplify(coeff(X(n), z, n))):
%p A075497_row := n -> op(PolynomialTools:-CoefficientList(Z(n), x)):
%p seq(A075497_row(n), n=0..9); # _Peter Luschny_, Jan 14 2018
%t Table[(2^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* _Michael De Vlieger_, Dec 31 2015 *)
%o (Sage) # uses[inverse_bell_transform from A265605]
%o multifact_2_2 = lambda n: prod(2*k + 2 for k in (0..n-1))
%o inverse_bell_matrix(multifact_2_2, 9) # _Peter Luschny_, Dec 31 2015
%o (PARI)
%o for(n=1, 11, for(m=1, n, print1(2^(n - m) * stirling(n, m, 2),", ");); print();) \\ _Indranil Ghosh_, Mar 25 2017
%Y Columns 1-7 are A000079, A006516, A016283, A025966, A075510-A075512.
%Y Row sums are A004211.
%Y Cf. A008277, A075498-A075505.
%K nonn,easy,tabl
%O 1,2
%A _Wolfdieter Lang_, Oct 02 2002