1,2

a(n,1)= A001715(n+2). a(n,m)=: S1p(4; 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).

The signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035469(n,m) := S2(4; 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+3 colors. The k roots (j=0) each come in one (or no) color. W. Lang, Oct 12 2007

W. Lang, First ten rows.

a(n, m) = n!*A030524(n, m)/(m!*3^(n-m)); a(n, m) = (3*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*(3-3*x+x^2)/(3*(1-x)^3))^m)/m!.

{1}; {4,1}; {20,12,1}; {120,128,24,1}; {840,1400,440,40,1}; E.g. Row polynomial E(3,x)= 20*x + 12*x^2 + x^3.

a(4,2)=128=4*(4*5)+3*(4*4) 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*4*5)=20 colored versions, e.g. ((1c1),(2c1,3c4,4c3)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 4 colors, c1..c4, can be chosen and the vertex labeled 4 with j=2 can come in 5 colors, e.g. c1..c5. Therefore there are 4*((1)*(1*4*5))=80 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*4)*(1*4))=48 such forests, e.g. ((1c1,3c2)(2c1,4c4)) or ((1c1,3c3)(2c1,4c2)), etc. W. Lang, Oct 12 2007

Cf. A049377 (row sums).

Alternating row sums A134137.

Sequence in context: A062137 A143497 A144354 this_sequence A144484 A121336 A126457

Adjacent sequences: A049349 A049350 A049351 this_sequence A049353 A049354 A049355

easy,nonn,tabl

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