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A049374 A triangle of numbers related to triangle A030527. 9
1, 6, 1, 42, 18, 1, 336, 276, 36, 1, 3024, 4200, 960, 60, 1, 30240, 66024, 23400, 2460, 90, 1, 332640, 1086624, 557424, 87360, 5250, 126, 1, 3991680, 18805248, 13349952, 2916144, 255360, 9912, 168, 1, 51891840, 342486144, 325854144, 95001984 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n,1) = A001725(n+4). a(n,m)=: S1p(6; 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).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A049385(n,m) =: S2(6; n,m). The monic row polynomials E(n,x) := Sum_{m=1..n} (a(n,m)*x^m), 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+5 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
LINKS
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
FORMULA
a(n, m) = n!*A030527(n, m)/(m!*5^(n-m)); a(n, m) = (5*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*(5 - 10*x + 10*x^2 - 5*x^3 + x^4)/(5*(1-x)^5))^m)/m!.
a(n,k) = n!* Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1) /(5^k*k!). - Vladimir Kruchinin, Apr 01 2011
EXAMPLE
Triangle begins
1;
6, 1;
42, 18, 1;
336, 276, 36, 1;
3024, 4200, 960, 60, 1;
30240, 66024, 23400, 2460, 90, 1;
332640, 1086624, 557424, 87360, 5250, 126, 1;
E.g., row polynomial E(3,x) = 42*x + 18*x^2 + x^3.
a(4,2) = 276 = 4*(6*7) + 3*(6*6) 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*6*7)=42 colored versions, e.g., ((1c1),(2c1,3c6,4c3)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 6 colors, c1..c6, can be chosen and the vertex labeled 4 with j=2 can come in 7 colors, e.g., c1..c7. Therefore there are 4*((1)*(1*6*7))=168 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*6)*(1*6))=108 such forests, e.g., ((1c1,3c4)(2c1,4c6)) or ((1c1,3c5)(2c1,4c2)), etc. - Wolfdieter Lang, Oct 12 2007
MAPLE
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> (n+5)!/120, 10); # Peter Luschny, Jan 28 2016
MATHEMATICA
a[n_, k_] = n!*Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n + 5j - 1, 5j - 1]/(5^k*k!), {j, 1, k}] ;
Flatten[Table[a[n, k], {n, 1, 9}, {k, 1, n}] ][[1 ;; 40]]
(* Jean-François Alcover, Jun 01 2011, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(#+5)!/120&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
PROG
(Maxima)
a(n, k)=(n!*sum((-1)^(k-j)*binomial(k, j)*binomial(n+5*j-1, 5*j-1), j, 1, k))/(5^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */
(PARI)
a(n, k)=(n!*sum(j=1, k, (-1)^(k-j)*binomial(k, j)*binomial(n+5*j-1, 5*j-1)))/(5^k*k!);
for(n=1, 12, for(k=1, n, print1(a(n, k), ", ")); print()); /* print triangle */ /* Joerg Arndt, Apr 01 2011 */
(GAP) Flat(List([1..10], n->Factorial(n)*List([1..n], k->Sum([1..k], j->(-1)^(k-j)*Binomial(k, j)*Binomial(n+5*j-1, 5*j-1)/(5^k*Factorial(k)))))); # Muniru A Asiru, Jun 23 2018
CROSSREFS
Cf. A049402 (row sums), A134140 (alternating row sums).
Sequence in context: A062138 A143498 A144356 * A283151 A138192 A136235
KEYWORD
easy,nonn,tabl
AUTHOR
STATUS
approved

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Last modified April 25 09:23 EDT 2024. Contains 371967 sequences. (Running on oeis4.)