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A134141 Generalized unsigned Stirling1 triangle, S1p(7). 4
1, 7, 1, 56, 21, 1, 504, 371, 42, 1, 5040, 6440, 1295, 70, 1, 55440, 114520, 36225, 3325, 105, 1, 665280, 2116800, 983920, 135975, 7105, 147, 1, 8648640, 40884480, 26714800, 5199145, 398860, 13426, 196, 1, 121080960, 826338240, 735469280 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A092082(n, m) =: S2(7; 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+6 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 05 2007

A triangle of numbers related to triangle A132166.

a(n,1)= A001730(n,5), n>=1. a(n,m)=: S1p(7; 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). S1p(3; n,m)= A046089(n,m), S1p(4; n,m)= A049352, S1p(5; n,m)= A049353(n,m), S1p(6; n,m)= A049374(n, m).

The Bell transform of factorial(n+6)/factorial(6). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

LINKS

Table of n, a(n) for n=1..39.

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

W. Lang, First ten rows.

FORMULA

a(n, m) = n!*A132166(n, m)/(m!*6^(n-m)); a(n, m) = (6*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*(6-15*x+20*x^2-15*x^3+6*x^4-x^5)/(6*(1-x)^6))^m)/m!.

EXAMPLE

{1}; {7,1}; {56,21,1}; {504,371,42,1}; ... E.g. Row polynomial E(3,x)=56*x+21*x^2+x^3.

a(4,2)= 371 = 4*(7*8)+3*(7*7) 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*7*8)=56 colored versions, e.g., ((1c1),(2c1,3c7,4c5)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 7 colors, c1..c7, can be chosen and the vertex labeled 4 with j=2 can come in 8 colors, e.g., c1..c8. Therefore there are 4*((1)*(1*7*8))=224 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*7)*(1*7))=147 such forests, e.g. ((1c1,3c4)(2c1,4c7)) or ((1c1,3c6)(2c1,4c2)), etc. - Wolfdieter Lang, Oct 05 2007

MAPLE

# The function BellMatrix is defined in A264428.

# Adds (1, 0, 0, 0, ..) as column 0.

BellMatrix(n -> (n+6)!/6!, 9); # Peter Luschny, Jan 27 2016

MATHEMATICA

a[n_, m_] /; n >= m >= 1 := a[n, m] = (6*m + 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}]][[1 ;; 39]] (* Jean-François Alcover, Jun 01 2011, after formula *)

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

rows = 12;

M = BellMatrix[(# + 6)!/6! &, rows];

Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

PROG

(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+6)/factorial(6), 10) # Peter Luschny, Jan 18 2016

CROSSREFS

First column A001730(n+5), n>=1.

Row sums A132164. Alternating row sums A132165.

Sequence in context: A052104 A144450 A051339 * A237111 A281620 A321187

Adjacent sequences:  A134138 A134139 A134140 * A134142 A134143 A134144

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Oct 12 2007

STATUS

approved

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Last modified March 20 07:27 EDT 2019. Contains 321345 sequences. (Running on oeis4.)