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A049352
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A triangle of numbers related to triangle A030524.
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11
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1, 4, 1, 20, 12, 1, 120, 128, 24, 1, 840, 1400, 440, 40, 1, 6720, 16240, 7560, 1120, 60, 1, 60480, 201600, 129640, 27720, 2380, 84, 1, 604800, 2681280, 2275840, 656320, 80080, 4480, 112, 1, 6652800, 38142720, 41370560, 15402240, 2498160, 196560
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
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LINKS
| W. Lang, First ten rows.
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FORMULA
| 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!.
a(n,k)=(n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+3*j-1,3*j-1)))/(3^k*k!) [From Vladimir Kruchinin kru(AT)tusur.ru Apr 01 2011]
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EXAMPLE
| {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
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PROG
| (Maxima) a(n, k):=(n!*sum((-1)^(k-j)*binomial(k, j)*binomial(n+3*j-1, 3*j-1), j, 1, k))/(3^k*k!); [From Vladimir Kruchinin kru(AT)tusur.ru Apr 01 2011]
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CROSSREFS
| Cf. A049377 (row sums).
Alternating row sums A134137.
Sequence in context: A062137 A143497 A144354 * A182867 A182826 A144484
Adjacent sequences: A049349 A049350 A049351 * A049353 A049354 A049355
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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