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A004747
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Triangle read by rows: the Bell transform of the triple factorial numbers A008544 without column 0.
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16
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1, 2, 1, 10, 6, 1, 80, 52, 12, 1, 880, 600, 160, 20, 1, 12320, 8680, 2520, 380, 30, 1, 209440, 151200, 46480, 7840, 770, 42, 1, 4188800, 3082240, 987840, 179760, 20160, 1400, 56, 1, 96342400, 71998080, 23826880, 4583040, 562800, 45360, 2352, 72, 1
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OFFSET
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1,2
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COMMENTS
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Previous name was: Triangle of numbers related to triangle A048966; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
T(n,m) = S2p(-2; n,m), a member of a sequence of triangles including S2p(-1; n,m) = A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) = A008277(n, m) (Stirling 2nd kind). T(n,1)= A008544(n-1).
T(n,m), n>=m>=1, enumerates unordered n-vertex m-forests composed of m plane (aka ordered) increasing (rooted) trees where vertices of out-degree r>=0 come in r+1 different types (like an (r+1)-ary vertex). Proof from the e.g.f. of the first column Y(z) = 1 - (1-3*x)^(1/3) and the F. Bergeron et al. eq. (8) Y'(z)= phi(Y(z)), Y(0) = 0, with out-degree o.g.f. phi(w)=1/(1-w)^2. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of the triple factorial numbers A008544 which adds a first column (1,0,0 ...) on the left side of the triangle. For the definition of the Bell transform see A264428. See A051141 for the triple factorial numbers A032031 and A203412 for the triple factorial numbers A007559 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 21 2015
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LINKS
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F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of increasing trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
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FORMULA
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T(n, m) = n!*A048966(n, m)/(m!*3^(n-m));
T(n+1, m) = (3*n-m)*T(n, m)+ T(n, m-1), for n >= m >= 1, with T(n, m) = 0, for n<m, and T(n, 0) = 0, T(1, 1) = 1.
E.g.f. of m-th column: ( 1 - (1-3*x)^(1/3) )^m/m!.
For a formula expressed as special values of hypergeometric functions 3F2 see the Maple program below. - Karol A. Penson, Feb 06 2004
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EXAMPLE
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Triangle begins:
1;
2, 1;
10, 6, 1;
80, 52, 12, 1;
880, 600, 160, 20, 1;
12320, 8680, 2520, 380, 30, 1;
209440, 151200, 46480, 7840, 770, 42, 1;
Tree combinatorics for T(3,2)=6: Consider first the unordered forest of m=2 plane trees with n=3 vertices, namely one vertex with out-degree r=0 (root) and two different trees with two vertices (one root with out-degree r=1 and a leaf with r=0). The 6 increasing labelings come then from the forest with rooted (x) trees x, o-x (1,(3,2)), (2,(3,1)) and (3,(2,1)) and similarly from the second forest x, x-o (1,(2,3)), (2,(1,3)) and (3,(1,2)).
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MAPLE
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T := (n, m) -> 3^n/m!*(1/3*m*GAMMA(n-1/3)*hypergeom([1-1/3*m, 2/3-1/3*m, 1/3-1/3*m], [2/3, 4/3-n], 1)/GAMMA(2/3)-1/6*m*(m-1)*GAMMA(n-2/3)*hypergeom( [1-1/3*m, 2/3-1/3*m, 4/3-1/3*m], [4/3, 5/3-n], 1)/Pi*3^(1/2)*GAMMA(2/3)):
for n from 1 to 6 do seq(simplify(T(n, k)), k=1..n) od;
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> mul(3*k+2, k=(0..n-1)), 9); # Peter Luschny, Jan 29 2016
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MATHEMATICA
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(* First program *)
T[1, 1]= 1; T[_, 0]= 0; T[0, _]= 0; T[n_, m_]:= (3*(n-1)-m)*T[n-1, m]+T[n-1, m-1];
Flatten[Table[T[n, m], {n, 12}, {m, n}] ][[1 ;; 45]] (* Jean-François Alcover, Jun 16 2011, after recurrence *)
(* Second program *)
f[n_, m_]:= m/n Sum[Binomial[k, n-m-k] 3^k (-1)^(n-m-k) Binomial[n+k-1, n-1], {k, 0, n-m}]; Table[n! f[n, m]/(m! 3^(n-m)), {n, 12}, {m, n}]//Flatten (* Michael De Vlieger, Dec 23 2015 *)
(* Third program *)
rows = 12;
T[n_, m_]:= BellY[n, m, Table[Product[3k+2, {k, 0, j-1}], {j, 0, rows}]];
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PROG
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(Sage) # uses [bell_transform from A264428]
triplefactorial = lambda n: prod(3*k+2 for k in (0..n-1))
trifact = [triplefactorial(k) for k in (0..n)]
return bell_transform(n, trifact)
(Magma)
if k eq 0 then return 0;
elif k eq n then return 1;
else return (3*(n-1)-k)*T(n-1, k) + T(n-1, k-1);
end if;
end function;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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