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A011801

Triangle read by rows, the inverse Bell transform of n!*binomial(4,n) (without column 0).

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)

OFFSET

1,2

COMMENTS

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).

For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

LINKS

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Peter Luschny, The Bell transform

Index entries for sequences related to Bessel functions or polynomials

FORMULA

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.

Sum_{k=1..n} T(n, k) = A028575(n).

EXAMPLE

Triangle starts:

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;

MATHEMATICA

(* 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;

Table[T[n, m], {n, 10}, {m, n}]//Flatten (* Jean-François Alcover, Jun 20 2018 *)

(* Alternative: *)

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;

A011801= Table[T[[n, m]], {n, rows}, {m, n}]//Flatten (* Jean-François Alcover, Jun 22 2018 *)

PROG

(SageMath) # 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)

function T(n, k) // T = A011801

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

CROSSREFS

Cf. A008277, A008546, A049223, A264428.

Cf. A028575 (row sums).

Triangles with the recurrence T(n,k) = (m*(n-1)-k)*T(n-1,k) + T(n-1,k-1): A010054 (m=1), A001497 (m=2), A004747 (m=3), A000369 (m=4), this sequence (m=5), A013988 (m=6).

Sequence in context: A217020 A329066 A144267 * A169656 A362589 A381102

Adjacent sequences: A011798 A011799 A011800 * A011802 A011803 A011804

KEYWORD

easy,nonn,tabl

AUTHOR

Wolfdieter Lang

EXTENSIONS

New name from Peter Luschny, Jan 16 2016

STATUS

approved