A013988 Triangle read by rows, the inverse Bell transform of n!*binomial(5,n) (without column 0).

1, 5, 1, 55, 15, 1, 935, 295, 30, 1, 21505, 7425, 925, 50, 1, 623645, 229405, 32400, 2225, 75, 1, 21827575, 8423415, 1298605, 103600, 4550, 105, 1, 894930575, 358764175, 59069010, 5235405, 271950, 8330, 140, 1, 42061737025, 17398082625, 3016869625, 289426830, 16929255, 621810, 14070, 180, 1

Offset

1,2

Comments

Previous name was: Triangle of numbers related to triangle A049224; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.

T(n, m) = S2p(-5; 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) = A008543(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.

Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3

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!*A049224(n, m)/(m!*6^(n-m));

T(n+1, m) = (6*n-m)*T(n, m) + T(n, m-1), for n >= m >= 1, with T(n, m) = 0, n<m, and T(n, 0) = 0, T(1, 1) = 1.

E.g.f. of m-th column: ((1 - (1-6*x)^(1/6))^m)/m!.

Sum_{k=1..n} T(n, k) = A028844(n).

Example

Triangle begins as:
          1;
          5,         1;
         55,        15,        1;
        935,       295,       30,       1;
      21505,      7425,      925,      50,      1;
     623645,    229405,    32400,    2225,     75,     1;
   21827575,   8423415,  1298605,  103600,   4550,   105,    1;
  894930575, 358764175, 59069010, 5235405, 271950,  8330,  140,   1;

Mathematica

(* First program *)
rows = 10;
b[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];
A = Table[b[n, m], {n, 1, rows}, {m, 1, rows}] // Inverse // Abs;
A013988 = Table[A[[n, m]], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==0, 0, If[k==n, 1, (6*(n-1) -k)*T[n-1, k] +T[n-1, k-1]]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 03 2023 *)

Prog

(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(5, n), 8) # Peter Luschny, Jan 16 2016
(Magma)
function T(n, k) // T = A013988
  if k eq 0 then return 0;
  elif k eq n then return 1;
  else return (6*(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, A008543, A049224, A264428.

Cf. A028844 (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), A011801 (m=5), this sequence (m=6).

Sequence in context: A144341 A144342 A144268 * A340472 A342318 A246006

Adjacent sequences: A013985 A013986 A013987 * A013989 A013990 A013991

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Extensions

New name from Peter Luschny, Jan 16 2016

Status

approved