Original entry on oeis.org
0, 1, 2, 4, 11, 47, 292, 2368, 23427, 272263, 3628872, 54525252, 911484163, 16775498551, 337021458884, 7338279413680, 172130372061035, 4327036966579151, 116046966039565672, 3307263639537314116
Offset: 0
a(x) = 0 + 1 + 1 + 2 + 7 + 36 + 245 + 2076 + 21059 + 248836 = 272263 is prime.
-
S:= rhs(dsolve({diff(a(x), x) = log(1/(1-a(x)))+1, a(0)=0}, a(x), series, order=31)):
L:= [seq(coeff(S, x, j)*j!, j=0..30)]:
ListTools:-PartialSums(L); # Robert Israel, Dec 21 2017
A008544
Triple factorial numbers: Product_{k=0..n-1} (3*k+2).
Original entry on oeis.org
1, 2, 10, 80, 880, 12320, 209440, 4188800, 96342400, 2504902400, 72642169600, 2324549427200, 81359229952000, 3091650738176000, 126757680265216000, 5577337931669504000, 262134882788466688000
Offset: 0
Joe Keane (jgk(AT)jgk.org)
a(2) = 10 from the described trees with 3 vertices: there are three trees with a root vertex (label 1) with outdegree r=2 (like the three 3-stars each with one different ray missing) and the four trees with a root (r=1 and label 1) a vertex with (r=1) and a leaf (r=0). Assigning labels 2 and 3 yields 2*3+4=10 such trees.
a(2) = 10. The 10 possible plane increasing trees on 3 vertices, where vertices of outdegree 1 come in 2 colors (denoted a or b) and vertices of outdegree 2 come in 3 colors (a, b or c), are:
.
1a 1b 1a 1b 1a 1b 1c
| | | | / \ / \ / \
2a 2b 2b 2a 2 3 2 3 2 3
| | | |
3 3 3 3 1a 1b 1c
/ \ / \ / \
3 2 3 2 3 2
- T. D. Noe, Table of n, a(n) for n = 0..100
- Murray Bremner and Vladimir Dotsenko, Associator dependent algebras and Koszul duality, arXiv:2203.11142 [math.RA], 2022.
- Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
- Keiichi Shigechi, On the lattice of weighted partitions, arXiv:2212.14666 [math.CO], 2022. See p. 27.
- Garrett Southwood and Hua Wang, The Statistics and Combinatorics of Increasing Trees and Colored Increasing Trees, J. Combin. Math. Combin. Comput. (2024) Vol. 123, 475-487. See p. 485.
-
a008544 n = a008544_list !! n
a008544_list = scanl (*) 1 a016789_list
-- Reinhard Zumkeller, Sep 20 2013
-
[Round((Gamma(2*n-5/3)/Gamma(n-5/6)*Gamma(2/3)/Gamma(5/6) )/ Sqrt(3)*3^n/4^(n-1)): n in [1..20]]; // Vincenzo Librandi, Feb 21 2015
-
[Round(3^n*Gamma(n+2/3)/Gamma(2/3)): n in [0..20]]; // G. C. Greubel, Mar 31 2019
-
a := n -> mul(3*k-1, k = 1..n);
A008544 := n -> mul(k, k = select(k-> k mod 3 = 2, [$1 .. 3*n])): seq(A008544(n), n = 0 .. 16); # Peter Luschny, Jun 23 2011
-
k = 3; b[1]=2; b[n_]:= b[n] = b[n-1]+k; a[0]=1; a[1]=2; a[n_]:= a[n] = a[n-1]*b[n]; Table[a[n], {n,0,20}] (* Roger L. Bagula, Sep 17 2008 *)
Product[3 k + 2, {k, 0, # - 1}] & /@ Range[0, 16] (* Michael De Vlieger, Jan 02 2016 *)
Table[3^n*Pochhammer[2/3, n], {n,0,20}] (* G. C. Greubel, Mar 31 2019 *)
-
a(n):=((n)!*sum(binomial(k,n-k)*binomial(n+k,k)*3^(-n+k)*(-1)^(n-k),k,floor(n/2),n)); /* Vladimir Kruchinin, Sep 28 2013 */
-
a(n) = prod(k=0,n-1, 3*k+2 );
-
vector(20, n, n--; round(3^n*gamma(n+2/3)/gamma(2/3))) \\ G. C. Greubel, Mar 31 2019
-
@CachedFunction
def A008544(n): return 1 if n == 0 else (3*n-1)*A008544(n-1)
[A008544(n) for n in (0..16)] # Peter Luschny, May 20 2013
-
[3^n*rising_factorial(2/3, n) for n in (0..20)] # G. C. Greubel, Mar 31 2019
A053492
REVEGF transform of [1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ...].
Original entry on oeis.org
1, 2, 15, 184, 3155, 69516, 1871583, 59542064, 2185497819, 90909876100, 4226300379983, 217152013181544, 12219893000227107, 747440554689309404, 49374719534173925055, 3503183373320829575008, 265693897270211120103563, 21451116469521758657525748
Offset: 1
E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3155*x^5/5! + ...
Related expansions from _Paul D. Hanna_, Jul 07 2012: (Start)
A(x) = x + (exp(x)-1)*x + d/dx (exp(x)-1)^2*x^2/2! + d^2/dx^2 (exp(x)-1)^3*x^3/3! + d^3/dx^3 (exp(x)-1)^4*x^4/4! + ...
log(A(x)/x) = (exp(x)-1) + d/dx (exp(x)-1)^2*x/2! + d^2/dx^2 (exp(x)-1)^3*x^2/3! + d^3/dx^3 (exp(x)-1)^4*x^3/4! + ... (End)
The a(3) = 15 pointed trees are 1[1 2[2 3]], 1[1 3[2 3]], 1[1[1 3] 2], 1[1[1 2] 3], 1[1 2 3], 2[1 2[2 3]], 2[1[1 3] 2], 2[2 3[1 3]], 2[2[1 2] 3], 2[1 2 3], 3[1 3[2 3]], 3[2 3[1 3]], 3[1[1 2] 3], 3[2[1 2] 3], 3[1 2 3].
-
A:= series(RootOf(exp(A053492:=%20n-%3E%20n!%20*%20coeff(A,%20x,%20n);%20%23%20_Brian%20Drake">Z)*_Z+x-2*_Z), x, 30): A053492:= n-> n! * coeff(A, x, n); # _Brian Drake, Apr 25 2006
-
Rest[CoefficientList[InverseSeries[Series[2*x-x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Oct 27 2014 *)
-
a(n):= if n=1 then 1 else sum(k!*stirling2(n-1,k)*binomial(n+k-1,n-1),k,1,n-1); /* Vladimir Kruchinin, May 10 2011 */
-
{a(n) = if( n<1, 0, n! * polcoeff( serreverse( 2*x - x * exp(x + x * O(x^n))), n))}; /* Michael Somos, Jun 06 2012 */
-
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^m*x^m/m!)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Jul 07 2012
for(n=1, 25, print1(a(n), ", "))
-
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)} \\ Paul D. Hanna, Jul 07 2012
for(n=1, 25, print1(a(n), ", "))
-
\p100 \\ set precision
{A=Vec(sum(n=0, 400, 1./(2 - n*x +O(x^25))^(n+1)) )}
for(n=1, #A, print1(round(A[n]), ", ")) \\ Paul D. Hanna, Oct 27 2014
A055356
Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 4, 2, 0, 1, 11, 18, 6, 0, 1, 26, 98, 96, 24, 0, 1, 57, 424, 874, 600, 120, 0, 1, 120, 1614, 6040, 8244, 4320, 720, 0, 1, 247, 5682, 35458, 83500, 83628, 35280, 5040, 0, 1, 502, 19022, 187288, 701164, 1169768, 915984, 322560, 40320, 0, 1
Offset: 1
Triangle begins
1;
1, 0;
1, 1, 0;
1, 4, 2, 0;
1, 11, 18, 6, 0;
1, 26, 98, 96, 24, 0;
1, 57, 424, 874, 600, 120, 0;
...
Alternating row sums give
A089963 (p(n+1,-1)).
-
P[1]:=1;for n from 1 to 8 do P[n+1]:=simplify((1+n*x)*P[n]+x*diff(P[n],x)) end; # F. Chapoton, Jul 16 2004
-
P[1][_] = 1;
P[n_][x_] := P[n][x] = (1 + (n-1) x) P[n-1][x] + x P[n-1]'[x] // Expand;
row[1] = {1};
row[n_] := Append[CoefficientList[P[n-1][x], x], 0];
Array[row, 10] // Flatten (* Jean-François Alcover, Nov 17 2018, after F. Chapoton *)
-
A(n)={my(v=vector(n)); v[1]=y; for(n=2, #v, v[n]=v[n-1] + sum(k=1, n-2, binomial(n-2, k)*v[k]*v[n-k])); vector(#v, i, Vecrev(v[i]/y, i))}
{ my(T=A(10)); for(i=1, #T, print(T[i])) } \\ Andrew Howroyd, Sep 23 2018
A032200
Number of rooted compound windmills (mobiles) of n nodes.
Original entry on oeis.org
1, 1, 2, 4, 9, 20, 51, 128, 345, 940, 2632, 7450, 21434, 62174, 182146, 537369, 1596133, 4767379, 14312919, 43162856, 130695821, 397184252, 1211057426, 3703794849, 11358759346, 34923477315, 107627138308, 332404636811
Offset: 1
From _Gus Wiseman_, Sep 05 2018: (Start)
The a(5) = 9 locally necklace plane trees:
((((o))))
(((oo)))
((o(o)))
(o((o)))
((o)(o))
((ooo))
(o(oo))
(oo(o))
(oooo)
(End)
- F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.84).
-
neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
neckplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[neckplane/@c],neckQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[neckplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)
-
CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+CIK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018
A038037
Number of labeled rooted compound windmills (mobiles) with n nodes.
Original entry on oeis.org
1, 2, 9, 68, 730, 10164, 173838, 3524688, 82627200, 2198295360, 65431163160, 2154106470240, 77714083773456, 3048821300491680, 129221979665461200, 5884296038166954240, 286492923374605966080, 14851359950834255500800
Offset: 1
- F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.83).
- Alois P. Heinz, Table of n, a(n) for n = 1..140
- C. G. Bower, Transforms (2)
- P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 454
- B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
- Index entries for sequences related to mobiles
-
logtr:= proc(p) local b; b:=proc(n) option remember; local k; if n=0 then 1 else p(n)- add(k *binomial(n,k) *p(n-k) *b(k), k=1..n-1)/n fi end end: b:= logtr(-a): a:= n-> `if`(n<=1,1, -n*b(n-1)): seq(a(n), n=1..25); # Alois P. Heinz, Sep 14 2008
-
a[n_] = Sum[Binomial[n, j]*Abs[StirlingS1[n-1, j]]*j!, {j, 0, n}]; Array[a, 18]
(* Jean-François Alcover, Jun 22 2011, after Vladimir Kruchinin *)
-
Vec(serlaplace(serreverse(x/(1 - log(1-x + O(x^20)))))) \\ Andrew Howroyd, Sep 19 2018
A007549
Number of increasing rooted connected graphs where every block is a complete graph.
Original entry on oeis.org
1, 1, 3, 14, 89, 716, 6967, 79524, 1041541, 15393100, 253377811, 4596600004, 91112351537, 1959073928124, 45414287553455, 1129046241331316, 29965290866974493, 845605519848379436, 25282324544244718411, 798348403914242674980, 26549922456617388029641
Offset: 1
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Vaclav Kotesovec, Table of n, a(n) for n = 1..410 (first 200 terms from Vincenzo Librandi)
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
-
exptr:= proc(p) local g; g:= proc(n) option remember; p(n) +add(binomial(n-1, k-1) *p(k) *g(n-k), k=1..n-1) end: end: b:= exptr(exptr(a)): a:= n-> `if`(n=0, 1, b(n-1)): seq(a(n), n=1..30); # Alois P. Heinz, Oct 07 2008
-
exptr[p_] := Module[{g}, g[n_] := g[n] = p[n] + Sum[ Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n-1}]; g]; b = exptr[ exptr[a] ]; a[n_] := If[n == 0, 1, b[n-1]]; Table[ a[n], {n, 1, 19}] (* Jean-François Alcover, May 10 2012, after Alois P. Heinz *)
A131178
Non-plane increasing unary binary (0-1-2) trees where the nodes of outdegree 1 come in 2 colors.
Original entry on oeis.org
1, 2, 5, 16, 64, 308, 1730, 11104, 80176, 643232, 5676560, 54650176, 569980384, 6401959328, 77042282000, 988949446144, 13488013248256, 194780492544512, 2969094574403840, 47640794742439936, 802644553810683904, 14166772337295285248, 261410917571703825920
Offset: 1
G.f. = x + 2*x^2 + 5*x^3 + 16*x^4 + 64*x^5 + 308*x^6 + 1730*x^7 + 11104*x^8 + ...
a(3) = 5: Denoting the two types of node of outdegree 1 by the letters a or b, the 5 possible trees are
.
. 1a 1b 1a 1b 1
. | | | | / \
. 2a 2b 2b 2a 2 3
. | | | |
. 3 3 3 3
- _Peter Bala_, Sep 01 2011
- Vincenzo Librandi, Table of n, a(n) for n = 1..100
- Jean-Luc Baril, Sergey Kirgizov, Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
- F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
- Lapo Cioni, Luca Ferrari, and Corentin Henriet. A direct bijection between two-stack sortable permutations and fighting fish, Euro. Conf. Comb., Graph Theory Appl. (2023) No. 12, 283-289.
- D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
-
E:= (2*(exp(sqrt(2)*x)-1)) / ((2+sqrt(2))-(2-sqrt(2))*exp(sqrt(2)*x)):
S:= map(simplify,series(E,x,101)):
seq(coeff(S,x,j)*j!, j=1..100); # Robert Israel, Nov 23 2016
-
max = 25; f[x_] := (2*(Exp[Sqrt[2]*x] - 1))/((2 + Sqrt[2]) - (2 - Sqrt[2])*Exp[Sqrt[2]*x]); Drop[ Simplify[ CoefficientList[ Series[f[x], {x, 0, max}], x]*Range[0, max]!], 1] (* Jean-François Alcover, Oct 05 2011 *)
-
x='x+O('x^66); /* that many terms */
default(realprecision,1000); /* working with floats here */
egf=(2*(exp(sqrt(2)*x)-1)) / ((2+sqrt(2))-(2-sqrt(2))*exp(sqrt(2)*x));
round(Vec(serlaplace(egf))) /* show terms */
/* Joerg Arndt, Sep 01 2011 */
-
/* the following program should be preferred. */
Vec( serlaplace( serreverse( intformal( 1/(1+2*x+1/2*x^2) + O(x^66) ) ) ) )
\\ Joerg Arndt, Mar 01 2014
-
{a(n) = if( n<1, 0, n! * polcoeff( 2 / (-2 + quadgen(8) * (-1 + 2 / (1 - exp(-quadgen(8) * x + x * O(x^n))))), n))};
Changed offset to 1 to agree with name and example. -
Michael Somos, Nov 23 2016
A108528
Number of increasing mobiles (cycle rooted trees) with n generators.
Original entry on oeis.org
1, 2, 10, 92, 1216, 20792, 435520, 10793792, 308874016, 10021509632, 363509706880, 14576530558592, 640275236943616, 30573223563625472, 1576805482203235840, 87353392124392020992, 5173324070004374358016, 326160898887563325581312, 21810458629345555407462400
Offset: 1
-
Rest[CoefficientList[InverseSeries[Series[Log[(1+x)*Sqrt[1-x^2]], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
-
{a(n)=n!*polcoeff(serreverse(log((1+x)*sqrt(1-x^2+O(x^(n+2))))),n)} \\ Paul D. Hanna, Sep 11 2010
A190015
Triangle T(n,k) for solving differential equation A'(x)=G(A(x)), G(0)!=0.
Original entry on oeis.org
1, 1, 2, 1, 6, 8, 1, 24, 42, 16, 22, 1, 120, 264, 180, 192, 136, 52, 1, 720, 1920, 1248, 540, 1824, 2304, 272, 732, 720, 114, 1, 5040, 15840, 10080, 8064, 18720, 22752, 9612, 7056, 10224, 17928, 3968, 2538, 3072, 240, 1, 40320, 146160, 92160, 70560, 32256, 207360, 249120, 193536, 73728, 61560, 144720, 246816, 101844, 142704, 7936, 51048, 110448, 34304, 8334, 11616, 494, 1
Offset: 0
Triangle begins:
1;
1;
2,1;
6,8,1;
24,42,16,22,1;
120,264,180,192,136,52,1;
720,1920,1248,540,1824,2304,272,732,720,114,1;
5040,15840,10080,8064,18720,22752,9612,7056,10224,17928,3968,2538,3072,240,1;
40320,146160,92160,70560,32256,207360,249120,193536,73728,61560,144720,246816, 101844,142704,7936,51048,110448,34304,8334,11616,494,1;
Example for n=5:
partitions of number 9 into 5 parts in lexicographic order:
[1,1,1,1,5]
[1,1,1,2,4]
[1,1,1,3,3]
[1,1,2,2,3]
[1,2,2,2,2]
a(5) = (24*g(0)^4*g(4) +42*g(0)^3*g(1)*g(3) +16*g(0)^3*g(2)^2 +22*g(0)^2*g(1)^2*g(2) +g(0)*g(1)^4)/5!.
-
/* array of triangle */
M:[1,1,2,1,6,8,1,24,42,16,22,1,120,264,180,192,136,52,1,720,1920,1248,540,1824,2304,272,732,720,114,1,5040,15840,10080,8064,18720,22752,9612,7056,10224,17928,3968,2538,3072,240,1,40320,146160,92160,70560,32256,207360,249120,193536,73728,61560,144720,246816,101844,142704,7936,51048,110448,34304,8334,11616,494,1];
/* function of triangle */
T(n,k):=M[sum(num_partitions(i),i,0,n-1)+k+1];
/* count number of partitions of n into m parts */
b(n,m):=if n
-
/* Find triangle */
Co(n,k):=if k=1 then a(n) else sum(a(i+1)*Co(n-i-1,k-1),i,0,n-k);
a(n):=if n=1 then 1 else 1/n*sum(Co(n-1,k)*x(k),k,1,n-1);
makelist(ratsimp(n!*a(n)),n,1,5);
/* Vladimir Kruchinin, Jun 15 2012 */
-
serlaplace( serreverse( intformal( 1 / sum(n=0, 9, eval(Str("g"n)) * x^n, x * O(x^9))))) /* Michael Somos, Oct 22 2014 */
Showing 1-10 of 13 results.
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