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

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn
• tabl

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

Examples

			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;
		

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

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).

Programs

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

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_] /; nJean-François Alcover, Jun 20 2018 *)
(* 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;
A011801= Table[T[[n, m]], {n,rows}, {m,n}]//Flatten (* Jean-François Alcover, Jun 22 2018 *)

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

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).

Extensions

New name from Peter Luschny, Jan 16 2016