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%I #33 Mar 28 2020 16:38:22
%S 1,7,1,56,21,1,504,371,42,1,5040,6440,1295,70,1,55440,114520,36225,
%T 3325,105,1,665280,2116800,983920,135975,7105,147,1,8648640,40884480,
%U 26714800,5199145,398860,13426,196,1,121080960,826338240,735469280
%N Generalized unsigned Stirling1 triangle, S1p(7).
%C 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).
%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+6 colors. The k roots (j=0) each come in one (or no) color. - _Wolfdieter Lang_, Oct 05 2007
%C A triangle of numbers related to triangle A132166.
%C 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).
%C The Bell transform of factorial(n+6)/factorial(6). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%H W. Lang, <a href="https://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 W. Lang, <a href="/A134141/a134141.txt">First ten rows</a>.
%F 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!.
%e {1}; {7,1}; {56,21,1}; {504,371,42,1}; ... E.g. Row polynomial E(3,x)=56*x+21*x^2+x^3.
%e 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
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0.
%p BellMatrix(n -> (n+6)!/6!, 9); # _Peter Luschny_, Jan 27 2016
%t 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 *)
%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t rows = 12;
%t M = BellMatrix[(# + 6)!/6! &, rows];
%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%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+6)/factorial(6), 10) # _Peter Luschny_, Jan 18 2016
%Y First column A001730(n+5), n>=1.
%Y Row sums A132164. Alternating row sums A132165.
%K nonn,easy,tabl
%O 1,2
%A _Wolfdieter Lang_, Oct 12 2007
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