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%I #35 Jun 02 2024 11:10:52
%S 1,4,1,20,12,1,120,128,24,1,840,1400,440,40,1,6720,16240,7560,1120,60,
%T 1,60480,201600,129640,27720,2380,84,1,604800,2681280,2275840,656320,
%U 80080,4480,112,1,6652800,38142720,41370560,15402240,2498160,196560
%N A triangle of numbers related to triangle A030524.
%C 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).
%C 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).
%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+3 colors. The k roots (j=0) each come in one (or no) color. - _Wolfdieter Lang_, Oct 12 2007
%C Also the Bell transform of A001715. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 28 2016
%H Wolfdieter Lang, <a href="/A049352/a049352.txt">First ten rows. </a>
%F 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!.
%F 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!). [_Vladimir Kruchinin_, Apr 01 2011]
%e Triangle starts:
%e {1};
%e {4,1};
%e {20,12,1};
%e {120,128,24,1};
%e {840,1400,440,40,1};
%e ...
%e E.g. Row polynomial E(3,x)= 20*x + 12*x^2 + x^3.
%e 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. - _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+3)!/6, 10); # _Peter Luschny_, Jan 28 2016
%t 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!); Table[a[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 26 2013, after _Vladimir Kruchinin_ *)
%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[(# + 3)!/6&, 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+3*j-1,3*j-1),j,1,k))/(3^k*k!); /* _Vladimir Kruchinin_, Apr 01 2011 */
%Y Cf. A049377 (row sums).
%Y Alternating row sums A134137.
%K easy,nonn,tabl
%O 1,2
%A _Wolfdieter Lang_
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