A046089 - OEIS

%I #67 Jan 04 2025 02:42:20

%S 1,3,1,12,9,1,60,75,18,1,360,660,255,30,1,2520,6300,3465,645,45,1,

%T 20160,65520,47880,12495,1365,63,1,181440,740880,687960,235305,35700,

%U 2562,84,1,1814400,9072000,10372320,4452840,877905,86940,4410,108,1

%N Triangle read by rows, the Bell transform of (n+2)!/2 without column 0.

%C Previous name was: A triangle of numbers related to triangle A030523.

%C 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).

%C 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).

%C 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

%C 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

%C 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

%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%H Wolfdieter Lang, <a href="/A046089/a046089.txt">First ten rows</a>.

%H E. Neuwirth, <a href="http://dx.doi.org/10.1016/S0012-365X(00)00373-3">Recursively defined combinatorial Functions: Extending Galton's board</a>, Discr. Maths. 239 (2001) 33-51.

%H John Riordan, <a href="/A002720/a002720_2.pdf">Letter, Apr 28 1976.</a>

%F 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!.

%F 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.

%F 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

%e Triangle begins:

%e   [1],

%e   [3, 1],

%e   [12, 9, 1],

%e   [60, 75, 18, 1],

%e   [360, 660, 255, 30, 1],

%e   [2520, 6300, 3465, 645, 45, 1],

%e   ...

%t 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 *)

%t 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_ *)

%t a[0] = 0; a[n_] := (n + 1)!/2;

%t 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}]];

%t Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 19 2016, after _Peter Luschny_, updated Jan 01 2021 *)

%t rows = 9;

%t a[n_, m_] := BellY[n, m, Table[(k+2)!/2, {k, 0, rows}]];

%t Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)

%o (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 */

%o (Sage) # uses[bell_matrix from A264428]

%o # Adds a column 1,0,0,0, ... at the left side of the triangle.

%o bell_matrix(lambda n: factorial(n+2)//2, 9) # _Peter Luschny_, Jan 19 2016

%Y Cf. A001710, A049376, A105752.

%Y Alternating row sums A134138.

%K easy,nonn,tabl,changed

%O 1,2

%A _Wolfdieter Lang_

%E New name from _Peter Luschny_, Jan 19 2016