OFFSET
1,2
COMMENTS
Previous name was: A triangle of numbers related to triangle A030523.
a(n,1)= A001710(n+1). a(n,m)=: S1p(3; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n,m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n,m) (unsigned Lah numbers).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035342(n,m) := S2(3; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+2 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
a(4,2)=75=4*(3*4)+3*(3*3) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*3*4)=12 colored versions, e.g. ((1c1),(2c1,3c3,4c2)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 3 colors, c1,c2 and c3, can be chosen and the vertex labeled 4 with j=2 can come in 4 colors, e.g. c1, c2, c3 and c4. Therefore there are 4*((1)*(1*3*4)=48 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*3)*(1*3))=27 such forests, e.g. ((1c1,3c2)(2c1,4c1)) or ((1c1,3c2)(2c1,4c2)), etc. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of A001710(n+2) (adding 1,0,0,.. as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016
LINKS
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First ten rows.
E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Maths. 239 (2001) 33-51.
John Riordan, Letter, Apr 28 1976.
FORMULA
a(n, m) = n!*A030523(n, m)/(m!*2^(n-m)); a(n, m) = (2*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n<m; a(n, 0) := 0; a(1, 1)=1. E.g.f. for m-th column: ((x*(2-x)/(2*(1-x)^2))^m)/m!.
a(n, m) = sum(|S1(n, j)|* A075497(j, m), j=m..n) (matrix product), with S1(n, j) := A008275(n, j) (signed Stirling1 triangle). Priv. comm. to Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference.
a(n, k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+2*j-1,2*j-1)))/(2^k*k!) - Vladimir Kruchinin, Apr 01 2011
EXAMPLE
Triangle begins:
[1],
[3, 1],
[12, 9, 1],
[60, 75, 18, 1],
[360, 660, 255, 30, 1],
[2520, 6300, 3465, 645, 45, 1],
...
MATHEMATICA
a[n_, m_] /; n >= m >= 1 := a[n, m] = (2m + n - 1)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* Jean-François Alcover, Jul 22 2011 *)
a[n_, k_] := -(-1/2)^k*(n+1)!*HypergeometricPFQ[{1-k, n/2+1, (n+3)/2}, {3/2, 2}, 1]/(k-1)!; Table[a[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 28 2013, after Vladimir Kruchinin *)
a[0] = 0; a[n_] := (n + 1)!/2;
T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, a[0]^n], Sum[Binomial[n - 1, j - 1] a[j] T[n - j, k - 1], {j, 0, n - k + 1}]];
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 19 2016, after Peter Luschny, updated Jan 01 2021 *)
rows = 9;
a[n_, m_] := BellY[n, m, Table[(k+2)!/2, {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)
PROG
(Maxima) a(n, k):=(n!*sum((-1)^(k-j)*binomial(k, j)*binomial(n+2*j-1, 2*j-1), j, 1, k))/(2^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */
(Sage) # uses[bell_matrix from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(lambda n: factorial(n+2)//2, 9) # Peter Luschny, Jan 19 2016
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
New name from Peter Luschny, Jan 19 2016
STATUS
approved