A049352 A triangle of numbers related to triangle A030524.
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
Offset: 1
Examples
Triangle starts:
{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. - _Wolfdieter Lang_, Oct 12 2007
Links
- Wolfdieter Lang, First ten rows.
Programs
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Maple
# The function BellMatrix is defined in A264428. # Adds (1,0,0,0, ..) as column 0. BellMatrix(n -> (n+3)!/6, 10); # Peter Luschny, Jan 28 2016
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Mathematica
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 *) 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[(# + 3)!/6&, rows]; Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *) -
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 */
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
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]
A043553 Row sums of convolution triangle A030524.
1, 7, 43, 250, 1420, 7990, 44800, 250900, 1404700, 7864000, 44026000, 246481000, 1379950000, 7725850000, 43254430000, 242167300000, 1355815600000, 7590769900000, 42498250000000, 237933871000000, 1332114307000000
Offset: 1
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Formula
G.f.: x*(1-3*x+3*x^2)/(1-10*x+30*x^2-30*x^3)= g1(4, x)/(1-g1(4, x)), g1(4, x) := x*(1-3*x+3*x^2)/(1-3*x)^3 (G.f. first column of A030524).
A036068 Expansion of (-1+1/(1-3*x)^3)/(9*x).
1, 6, 30, 135, 567, 2268, 8748, 32805, 120285, 433026, 1535274, 5373459, 18600435, 63772920, 216827928, 731794257, 2453663097, 8178876990, 27119434230, 89494132959, 294052151151, 962352494676, 3138105960900, 10198844372925
Offset: 0
Comments
G.f. for a(n)=A027472(n+3), n >= 0, is 1/(1-3*x)^3.
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- Index entries for linear recurrences with constant coefficients, signature (9, -27, 27).
Programs
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Mathematica
CoefficientList[Series[((1/(1-3x))^3-1)/(9x),{x,0,30}],x] (* Harvey P. Dale, Nov 26 2018 *)
Formula
a(n) = 3^(n-1)*binomial(n+3, 2); G.f.: (-1+(1-3*x)^(-3))/(x*3^2)=(1-3*x+3*x^2)/(1-3*x)^3.
G.f.: F(4,1;2;3x); [From Paul Barry, Sep 03 2008]
D-finite with recurrence: (n+1)*a(n) +3*(-n-3)*a(n-1)=0. - R. J. Mathar, Jan 28 2020
Comments