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A046089 Triangle read by rows, the Bell transform of (n+2)!/2 without column 0. 12
1, 3, 1, 12, 9, 1, 60, 75, 18, 1, 360, 660, 255, 30, 1, 2520, 6300, 3465, 645, 45, 1, 20160, 65520, 47880, 12495, 1365, 63, 1, 181440, 740880, 687960, 235305, 35700, 2562, 84, 1, 1814400, 9072000, 10372320, 4452840, 877905, 86940, 4410, 108, 1 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..45.

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

W. 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 *)

Unprotect[Power]; 0^0=1; a[0]=0; a[n_] := (n+1)!/2; T[n_, k_] := T[n, k] = If[k==0, 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 *)

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)

# The function bell_matrix is defined in 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

Cf. A001710, A049376, A105752.

Alternating row sums A134138.

Sequence in context: A156366 A144353 A039811 * A113360 A089434 A268298

Adjacent sequences:  A046086 A046087 A046088 * A046090 A046091 A046092

KEYWORD

easy,nonn,tabl

AUTHOR

Wolfdieter Lang

EXTENSIONS

New name from Peter Luschny, Jan 19 2016

STATUS

approved

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Last modified March 26 12:43 EDT 2019. Contains 321497 sequences. (Running on oeis4.)