A049353 - OEIS

%I #47 Aug 28 2019 16:41:51

%S 1,5,1,30,15,1,210,195,30,1,1680,2550,675,50,1,15120,34830,14025,1725,

%T 75,1,151200,502740,287280,51975,3675,105,1,1663200,7692300,5961060,

%U 1482705,151200,6930,140,1,19958400,124740000,126913500,41545980

%N A triangle of numbers related to triangle A030526.

%C 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).

%C 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).

%C 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. - _Wolfdieter Lang_, Oct 12 2007

%C Also the Bell transform of A001720. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 28 2016

%H W. Lang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%H W. Lang, <a href="/A049353/a049353.txt">First ten rows. </a>

%F 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!.

%F 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!). - _Vladimir Kruchinin_, Apr 01 2011

%e Triangle begins:

%e   {1};

%e   {5,1};

%e   {30,15,1}; E.g., row polynomial E(3,x)=30*x+15*x^2+x^3.

%e   {210,195,30,1};

%e   ...

%e 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. - _Wolfdieter Lang_, Oct 12 2007

%p # The function BellMatrix is defined in A264428.

%p # Adds (1,0,0,0, ..) as column 0.

%p BellMatrix(n -> (n+4)!/24, 10); # _Peter Luschny_, Jan 28 2016

%t 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}]] (* _Jean-François Alcover_, Jul 22 2011 *)

%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];

%t rows = 10;

%t M = BellMatrix[(#+4)!/24&, rows];

%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)

%o (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!); /* _Vladimir Kruchinin_, Apr 01 2011 */

%Y Cf. A049378 (row sums).

%Y Cf. A134139 (alternating row sums).

%K easy,nonn,tabl

%O 1,2

%A _Wolfdieter Lang_