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A011801
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Triangle read by rows, the inverse Bell transform of n!*binomial(4,n) (without column 0).
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13
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1, 4, 1, 36, 12, 1, 504, 192, 24, 1, 9576, 3960, 600, 40, 1, 229824, 100656, 17160, 1440, 60, 1, 6664896, 3048192, 563976, 54600, 2940, 84, 1, 226606464, 107255232, 21095424, 2256576, 142800, 5376, 112, 1, 8837652096, 4302305280, 887785920, 102332160, 7254576, 325584, 9072, 144, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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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 A049223; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
T(n, m) = S2p(-4; 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) = A008546(n-1).
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LINKS
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FORMULA
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T(n, m) = n!*A049223(n, m)/(m!*5^(n-m)).
T(n+1, m) = (5*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 n-th column: (1/n!)*( 1 - (1-5*x)^(1/5) )^n.
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EXAMPLE
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Triangle starts:
1;
4, 1;
36, 12, 1;
504, 192, 24, 1;
9576, 3960, 600, 40, 1;
229824, 100656, 17160, 1440, 60, 1;
6664896, 3048192, 563976, 54600, 2940, 84, 1;
226606464, 107255232, 21095424, 2256576, 142800, 5376, 112, 1;
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MATHEMATICA
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(* First program *)
T[n_, m_] /; n>=m>=1:= T[n, m]= (5*(n-1)-m)*T[n-1, m] + T[n-1, m-1]; T[n_, m_] /; n<m=0; T[_, 0]=0; T[1, 1]=1;
(* Second program *)
rows = 10;
b[n_, m_]:= BellY[n, m, Table[k! Binomial[4, k], {k, 0, rows}]];
T= Table[b[n, m], {n, rows}, {m, rows}]//Inverse//Abs;
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PROG
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(Sage) # uses[inverse_bell_matrix from A264428]
# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
inverse_bell_matrix(lambda n: factorial(n)*binomial(4, n), 8) # Peter Luschny, Jan 16 2016
(Magma)
if k eq 0 then return 0;
elif k eq n then return 1;
else return (5*(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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