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A075497
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Stirling2 triangle with scaled diagonals (powers of 2).
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16
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1, 2, 1, 4, 6, 1, 8, 28, 12, 1, 16, 120, 100, 20, 1, 32, 496, 720, 260, 30, 1, 64, 2016, 4816, 2800, 560, 42, 1, 128, 8128, 30912, 27216, 8400, 1064, 56, 1, 256, 32640, 193600, 248640, 111216, 21168, 1848, 72, 1
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OFFSET
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1,2
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COMMENTS
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This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
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.
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
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
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LINKS
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FORMULA
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a(n, m) = (2^(n-m)) * Stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*2)^(n-m))/(m-1)! for n >= m >= 1, else 0.
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.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-2*k*x), m >= 1.
E.g.f. for m-th column: (((exp(2*x)-1)/2)^m)/m!, m >= 1.
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
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.
R(n+1,x)/x = (x + 2) o (x + 2) o...o (x + 2) (n factors).
R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*2^(n-k)*R(k,x).
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)
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EXAMPLE
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[1];[2,1];[4,6,1]; ...; p(3,x) = x*(4 + 6*x + x^2).
Triangle (0, 2, 0, 4, 0, 6, 0, 8, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
1
0, 1
0, 2, 1
0, 4, 6, 1
0, 8, 28, 12, 1
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MAPLE
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with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(
binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
# Alternatively, giving the triangle in the form displayed in the Example section:
gf := exp(x*exp(z)*sinh(z)):
X := n -> series(gf, z, n+2):
Z := n -> n!*expand(simplify(coeff(X(n), z, n))):
A075497_row := n -> op(PolynomialTools:-CoefficientList(Z(n), x)):
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MATHEMATICA
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Table[(2^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)
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PROG
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(Sage) # uses[inverse_bell_transform from A265605]
multifact_2_2 = lambda n: prod(2*k + 2 for k in (0..n-1))
inverse_bell_matrix(multifact_2_2, 9) # Peter Luschny, Dec 31 2015
(PARI)
for(n=1, 11, for(m=1, n, print1(2^(n - m) * stirling(n, m, 2), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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