A049353 A triangle of numbers related to triangle A030526.
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
Offset: 1
Examples
Triangle begins:
{1};
{5,1};
{30,15,1}; E.g., row polynomial E(3,x)=30*x+15*x^2+x^3.
{210,195,30,1};
...
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
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- W. 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+4)!/24, 10); # Peter Luschny, Jan 28 2016
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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}]] (* _Jean-François Alcover, Jul 22 2011 *) 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[(#+4)!/24&, rows]; Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
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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 */
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
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
A045624 Row sums of convolution triangle A030526.
1, 11, 101, 851, 6885, 54723, 432021, 3403859, 26811397, 211225187, 1664405621, 13116776819, 103376383461, 814752361347, 6421443995733, 50610420076691, 398884119723973, 3143787312038051, 24777605586822197, 195283435452156851
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Wolfdieter 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 (17,-102,272,-272).
Programs
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GAP
a:=[1,11,101,851];; for n in [5..40] do a[n]:=17*a[n-1]-102*a[n-2] +272*a[n-3]-272*a[n-4]; od; a; # G. C. Greubel, Jan 13 2020
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Magma
R
:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1-6*x+16*x^2-16*x^3)/(1-17*x+102*x^2-272*x^3 + 272*x^4) )); // G. C. Greubel, Jan 13 2020 -
Maple
seq(coeff(series(x*(1-6*x+16*x^2-16*x^3)/(1-17*x+102*x^2-272*x^3 + 272*x^4), x, n+1), x, n), n = 1..40); # G. C. Greubel, Jan 13 2020
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Mathematica
Rest@CoefficientList[Series[x*(1-6*x+16*x^2-16*x^3)/(1-17*x+102*x^2-272*x^3 + 272*x^4), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *) -
PARI
my(x='x+O('x^40)); Vec(x*(1-6*x+16*x^2-16*x^3)/(1-17*x+102*x^2-272*x^3 + 272*x^4)) \\ G. C. Greubel, Jan 13 2020 -
Sage
def A045624_list(prec): P.
= PowerSeriesRing(ZZ, prec) return P( x*(1-6*x+16*x^2-16*x^3)/(1-17*x+102*x^2-272*x^3 + 272*x^4) ).list() a=A045624_list(40); a[1:] # G. C. Greubel, Jan 13 2020
Formula
G.f.: x*(1 -6*x +16*x^2 -16*x^3)/(1 -17*x +102*x^2 -272*x^3 +272*x^4) = g1(5, x)/(1-g1(5, x)), g1(5, x) := x*(1-6*x+16*x^2-16*x^3)/(1-4*x)^4 (G.f. first column of A030526).
A036070 Expansion of (-1+1/(1-4*x)^4)/(16*x); related to A038846.
1, 10, 80, 560, 3584, 21504, 122880, 675840, 3604480, 18743296, 95420416, 477102080, 2348810240, 11408506880, 54760833024, 260113956864, 1224065679360, 5712306503680, 26456998543360, 121702193299456
Offset: 0
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 (16, -96, 256, -256).
Formula
a(n) = 4^(n-1)*binomial(n+4, 3); G.f. (-1+(1-4*x)^(-4))/(x*4^2).
A132166 A convolution triangle of numbers obtained from A036224.
1, 21, 1, 336, 42, 1, 4536, 1113, 63, 1, 54432, 23184, 2331, 84, 1, 598752, 412272, 65205, 3990, 105, 1, 6158592, 6531840, 1518048, 139860, 6090, 126, 1, 60046272, 94618368, 30912840, 4010769, 256410, 8631, 147, 1, 560431872, 1274921856
Offset: 1
Comments
Examples
{1};{21,1};{336,42,1};{4536,1113,63,1};...; Row polynomial s(3,x)=336*x+42*x^2+x^3.
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- W. Lang, First ten rows.
Crossrefs
Formula
a(n, m) = 6*(6*m+n-1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n
G.f. for m-th column: ((1-(1-6*x)^6)/(36*(1-6*x)^6))^m.
Comments