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A049353
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A triangle of numbers related to triangle A030526.
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10
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1, 5, 1, 30, 15, 1, 210, 195, 30, 1, 1680, 2550, 675, 50, 1, 15120, 34830, 14025, 1725, 75, 1, 151200, 502740, 287280, 51975, 3675, 105, 1, 1663200, 7692300, 5961060, 1482705, 151200, 6930, 140, 1, 19958400, 124740000, 126913500, 41545980
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
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LINKS
| W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.
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FORMULA
| 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!.
a(n,k)=(n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+4*j-1,4*j-1)))/(4^k*k!) [From Vladimir Kruchinin kru(AT)tusur.ru Apr 1 2011]
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EXAMPLE
| {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
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MATHEMATICA
| a[n_, m_] /; n >= m >= 1 := a[n, m] = (4m + 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}]] (* From Jean-François Alcover, Jul 22 2011 *)
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PROG
| (Maxima) a(n, k):=(n!*sum((-1)^(k-j)*binomial(k, j)*binomial(n+4*j-1, 4*j-1), j, 1, k))/(4^k*k!); [From Vladimir Kruchinin kru(AT)tusur.ru Apr 1 2011]
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CROSSREFS
| Cf. A049378 (row sums).
Cf. A134139 (alternating row sums).
Sequence in context: A145926 A062140 A144355 * A165226 A027759 A197654
Adjacent sequences: A049350 A049351 A049352 * A049354 A049355 A049356
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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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