1,2

a(n,1)= A001720(n+3). a(n,m)=: S1p(5; 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(n,m).

Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A049029(n,m) := S2(5; 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+4 colors. The k roots (j=0) each come in one (or no) color. W. Lang, Oct 12 2007

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

W. Lang, First ten rows.

a(n, m) = n!*A030526(n, m)/(m!*4^(n-m)); a(n, m) = (4*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)*(2-2*x+x^2)/(4*(1-x)^4))^m)/m!.

{1}; {5,1}; {30,15,1}; {210,195,30,1}; ... E.g. Row polynomial E(3,x)=30*x+15*x^2+x^3.

a(4,2)= 195 =4*(5*6)+3*(5*5) 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*5*6)=30 colored versions, e.g. ((1c1),(2c1,3c5,4c6)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 5 colors, c1..c5, can be chosen and the vertex labeled 4 with j=2 can come in 6 colors, e.g. c1..c6. Therefore there are 4*((1)*(1*5*6))=120 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*5)*(1*5))=75 such forests, e.g. ((1c1,3c4)(2c1,4c5)) or ((1c1,3c5)(2c1,4c2)), etc. W. Lang, Oct 12 2007

Cf. A049378 (row sums).

Cf. A134139 (alternating row sums).

Sequence in context: A145926 A062140 A144355 this_sequence A165226 A027759 A066833

Adjacent sequences: A049350 A049351 A049352 this_sequence A049354 A049355 A049356

easy,nonn,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)