A124212 -id:A124212 - cp's OEIS frontend

A124214 E.g.f.: exp(x) / (2 - exp(3*x))^(1/3).

1, 2, 10, 98, 1402, 26162, 601930, 16462658, 521659162, 18791451602, 758345497450, 33889063202018, 1661229537252922, 88627461127536242, 5112116659677605770, 317007674364657538178, 21030558126242472270682

Offset: 0

Karol A. Penson, Oct 19 2006

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A229558, A004987, A124212.

A124214 := proc(n)
    exp(x)/root[3](2-exp(3*x)) ;
    coeftayl(%,x=0,n)*n! ;
end proc: # R. J. Mathar, Dec 19 2013

CoefficientList[Series[Exp[x]/(2-Exp[3*x])^(1/3), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A+A^4+x*O(x^n))); n!*polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 18 2013

a(n) ~ Gamma(2/3)*3^(n+1/2)*n^(n-1/6)/(sqrt(2*Pi)*exp(n)*(log(2))^(n+1/3)). - Vaclav Kotesovec, Jun 26 2013
E.g.f. A(x) satisfies: A'(x) = A(x) + A(x)^4. - Paul D. Hanna, Dec 18 2013
E.g.f. A(x) satisfies: A(x) = exp(x + Integral A(x)^3 dx) with A(0)=1. - Paul D. Hanna, Dec 18 2013
a(n) = 2^(-1/3) * Sum_{k >= 0} (1/18)^k*A004987(k)*(3*k + 1)^n = 2^(-1/3) * Sum_{k >= 0} (-1/2)^k*binomial(-1/3, k)*(3*k + 1)^n. Cf. A124212 and A229558. - Peter Bala, Aug 30 2016

A186695 A Galton triangle: T(n,k) = (2k-1)*(T(n-1,k) + T(n-1,k-1)): a type B analog of the ordered Bell numbers A019538.

Original entry on oeis.org

1, 1, 3, 1, 12, 15, 1, 39, 135, 105, 1, 120, 870, 1680, 945, 1, 363, 4950, 17850, 23625, 10395, 1, 1092, 26565, 159600, 373275, 374220, 135135, 1, 3279, 138285, 1303155, 4795875, 8222445, 6621615, 2027025

Offset: 1

Peter Bala, Mar 26 2011

The row polynomials R(n,x) of A019538 satisfy the recurrence relation R(n+1,x) = x*d/dx((1+x)*R(n,x)), and have the expansion R(n,x) = Sum_{k = 1..n} k!*Stirling2(n,k)*x^k.
Here we consider a sequence of polynomials P(n,x) (n>=1) defined by means of the similar recursion P(n+1,x) = x*d/dx((1+x^2)*P(n,x)), with starting value P(1,x) = x.
The first few polynomials are P(1,x) = x, P(2,x) = x + 3*x^3, P(3,x) = x + 12*x^3 + 15*x^5, and P(4,x) = x + 39*x^3 + 135*x^5 + 105*x^7.
Clearly, the P(n,x) are odd polynomials of the form P(n,x) = Sum_{k = 1..n} T(n,k)*x^(2*k-1).
This triangle lists the coefficients T(n,k). They are related to A039755, the type B Stirling numbers of Suter, by T(n,k) = (2*k-1)!!*A039755(n-1,k-1).

			Triangle begins
  n\k.|..1.....2.....3......4......5......6
  =========================================
  ..1.|..1
  ..2.|..1.....3
  ..3.|..1....12....15
  ..4.|..1....39...135....105
  ..5.|..1...120...870...1680....945
  ..6.|..1...363..4950..17850..23625..10395
  ..
Examples of recurrence relation
  T(4,3) = 5*(T(3,3)+T(3,2)) = 5*(15+12) = 135;
  T(6,4) = 7*(T(5,4)+T(5,3)) = 7*(1680+870) = 17850.

Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Tech Report TR 99-05, 1999.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).

Cf. A001147, A019538, A039755, A124212, A145901, A156919, A187075.

A186695 := proc(n, k) option remember; if k < 1 or k > n then 0; elif k = 1 then 1; else (2*k-1)*(procname(n-1, k) + procname(n-1, k-1)) ; end if; end proc: seq(seq(A186695(n,k),k = 1..n),n = 1..10);

T[n_, k_] := (2k-1)! Sum[(-1)^(k-j-1) (2j+1)^(n-1) Binomial[k-1, j], {j, 0, k-1}] / (2^(k-1) (k-1)!)^2;
Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 02 2019 *)

T(n+1,k+1) = ((2*k+1)!/(2^k*k!)^2)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(2*j+1)^n.
Recurrence relation: T(n,k) = (2k-1)*(T(n-1,k) + T(n-1,k-1)) with boundary conditions T(n,1) = 1, T(1,k) = 0 for k >= 2.
E.g.f.: F(x,t) = (x/(1+x))*(exp(t)/sqrt((1+x) - x*exp(2*t)) - 1) = Sum_{n>=1} R(n,x)*t^n/n! = x*t + (x + 3*x^2)*t^2/2! + (x + 12*x^2 + 15*x^3)*t^3/3! + ....
Compare with the e.g.f. for A019538, which is (x/(1+x))*(exp(t)/((1+x) - x*exp(t))-1).
The row polynomials R(n,x) are related to the polynomials P(n,x) of the comments section by P(n,x) = 1/x*R(n,x^2).
The generating function F(x,t) satisfies the partial differential equation d/dt(F) = 2*x*(1+x)*d/dx(F) + (x-1)*F + x.
It follows that the polynomials P(n,x) := Sum_{k = 1..n} T(n,k)*x^(2*k-1) satisfy the recurrence P(n+1,x) = x*d/dx((1+x^2)*P(n,x)), with P(1,x) = x. (Cf. the recurrence relation for row polynomials of A185896.)
The recurrence relation for T(n,k) given above now follows.
The row polynomials R(n,x) = Sum_{k = 1..n} T(n,k)*x^k satisfy R(n,-x-1) = (-1)^n*((1+x)/x)*S(n,x), where S(n,x) is the n-th row polynomial of A187075.
In addition, R(n,1/(x-1)) = (1/(x-1)^n)*Q(n-1,x), where Q(n,x) is the n-th row polynomial of A156919.
Row sums are [1,4,28,280,3616...] = 1/2*A124212(n) for n >= 1.
Main diagonal is [1,3,15,105,...] = A001147(k) for k >= 1.
Put S(n) = sum {k = 1..n} (-1)^k*T(n,k)/(k+1). Then for m>=2, S(2*m-1) = S(2*m) = (4^m-1)*Bernoulli(2*m)/m.
From Peter Bala, Aug 30 2016: (Start)
n-th row polynomial R(n,x) = 1/(1 + x)^(3/2) * Sum_{k >= 0} (1/4)^k*(x/(1 + x))^k*binomial(2*k,k)*(2*k + 1)^n.
R(n,x) = (1/(1 + x))*Sum_{k = 0..n} binomial(2*k,k)*A145901(n,k)*(x/4)^k. (End)

A229558 E.g.f.: exp(x) / (2 - exp(4*x))^(1/4).

Original entry on oeis.org

1, 2, 12, 152, 2832, 69152, 2089152, 75204992, 3142025472, 149428961792, 7969790856192, 471098477484032, 30567292903821312, 2159857294035525632, 165083372031671058432, 13570774387950150582272, 1193933787763434969956352, 111932230270819401046556672

Offset: 0

Paul D. Hanna, Dec 18 2013

			E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 152*x^3/3! + 2832*x^4/4! + 69152*x^5/5! +...
where A(x)^5 = 1 + 10*x + 140*x^2/2! + 2680*x^3/3! + 66320*x^4/4! +...
Also, A(x)^4 = 1 + 8*x + 96*x^2/2! + 1664*x^3/3! + 38400*x^4/4! +...
and log(A(x)) = 2*x + 8*x^2/2! + 96*x^3/3! + 1664*x^4/4! + 38400*x^5/5! +...

G. C. Greubel, Table of n, a(n) for n = 0..345

Cf. A124214, A034385, A124212.

CoefficientList[Series[E^x/(2-E^(4*x))^(1/4), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Dec 19 2013 *)

{a(n)=local(A=1+x,X=x+x*O(x^n));n!*polcoeff(exp(X)/(2-exp(4*X))^(1/4),n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A+A^5+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f. A(x) satisfies: A'(x) = A(x) + A(x)^5.
E.g.f. A(x) satisfies: A(x) = exp(x + Integral A(x)^4 dx).
a(n) ~ GAMMA(3/4) * 4^n * n^(n-1/4) / (sqrt(Pi) * exp(n) * log(2)^(n+1/4)). - Vaclav Kotesovec, Dec 19 2013
a(n) = 1/2^(1/4) * Sum_{k >= 0} (1/32)^k*A034385(k)*(4*k + 1)^n = 1/2^(1/4)*Sum_{k >= 0} (-1/2)^k*binomial(-1/4, k)*(4*k + 1)^n. Cf. A124212 and A124214. - Peter Bala, Aug 30 2016

A276371 Expansion of e.g.f. exp(x/2)/(2 - exp(2*x))^(1/4).

Original entry on oeis.org

1, 1, 3, 19, 177, 2161, 32643, 587539, 12273537, 291853441, 7782998883, 230028553459, 7462717994097, 263654454838321, 10075889406229923, 414147167601017779, 18217983822073897857, 853975145498805244801, 42495107452208870429763, 2237264405984004517212499, 124243242448367338311920817, 7258224393227482972980320881, 444967879322677755285771182403, 28563002475012109334240250609619

Offset: 0

Paul D. Hanna, Sep 09 2016

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 177*x^4/4! + 2161*x^5/5! + 32643*x^6/6! + 587539*x^7/7! + 12273537*x^8/8! + 291853441*x^9/9! + 7782998883*x^10/10! +...
such that A(x) = exp(x/2)/(2 - exp(2*x))^(1/4).

Paul D. Hanna, Table of n, a(n) for n = 0..520

Cf. A124212.

With[{nn = 50}, CoefficientList[Series[Exp[x/2]/(2 - Exp[2*x])^(1/4), {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, Apr 09 2017 *)

{a(n)=local(A=1+x, X=x+x*O(x^n)); A = exp(X/2)/(2-exp(2*X))^(1/4); n!*polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

E.g.f. A(x) satisfies: A'(x) = A(x)*(1 + A(x)^4)/2 with A(0)=1.
a(2*n) = 0 (mod 3), a(2*n+1) = 1 (mod 3), for n>=0.
a(n) ~ Gamma(3/4) * 2^n * n^(n-1/4) / (sqrt(Pi) * exp(n) * (log(2))^(n+1/4)). - Vaclav Kotesovec, Sep 11 2016
From Seiichi Manyama, Nov 16 2023: (Start)
a(n) = Sum_{k=0..n} (-2)^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-2)^k * (3/2 * k/n - 2) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n-1} 2^k * binomial(n-1,k) * a(n-k). (End)

A124213 Expansion of e.g.f.: exp(exp(x)/sqrt(2-exp(2*x))-1).

Original entry on oeis.org

1, 2, 12, 112, 1408, 22144, 417216, 9148416, 228649472, 6412193280, 199301663744, 6798026395648, 252397715738624, 10131114555244544, 437100940892913664, 20169428831476678656, 991081906535967948800, 51662621871173444698112, 2847287574653833612623872

Offset: 0

Karol A. Penson, Oct 19 2006

nonn

Exponential transform of A124212.

G. C. Greubel, Table of n, a(n) for n = 0..380

Cf. A124212.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(x)/Sqrt(2-Exp(2*x))-1))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 27 2018

With[{nmax = 50}, CoefficientList[Series[Exp[Exp[x]/Sqrt[2 - Exp[2*x]] - 1], {x,0,nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 27 2018 *)

x='x+O('x^30); Vec(serlaplace(exp(exp(x)/sqrt(2-exp(2*x))-1))) \\ G. C. Greubel, Sep 27 2018

E.g.f.: exp(exp(x)/sqrt(2-exp(2*x))-1).

a(n) ~ 2^(n + 1/6) * exp(3*n^(1/3)/(2^(2/3) * log(2)^(1/3)) - n - 1) * n^(n - 1/3) / (sqrt(3) * log(2)^(n + 1/6)). - Vaclav Kotesovec, Jun 26 2022

A365777 Expansion of e.g.f. (exp(2x) / (2 - exp(2x)))^(3/4).

Original entry on oeis.org

1, 3, 15, 117, 1257, 17163, 284055, 5522877, 123344817, 3111071283, 87454712895, 2710961144037, 91862770847577, 3378032307195003, 133970268354806535, 5699864583381903597, 258956671286986317537, 12512342291081486212323, 640686944845321006836975

Offset: 0

Seiichi Manyama, Nov 16 2023

nonn

Cf. A124212, A276371.

Cf. A365794.

a(n) = sum(k=0, n, (-2)^(n-k)*prod(j=0, k-1, 4*j+3)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (-2)^(n-k) * (Product_{j=0..k-1} (4*j+3)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-2)^k * (1/2 * k/n - 2) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3*a(n-1) + Sum_{k=1..n-1} 2^k * binomial(n-1,k) * a(n-k).

A176785 Sequence with e.g.f. g(x) = -(1/2)sqrt(2exp(-2*x)-1) + 1/2.

Original entry on oeis.org

0, 1, 0, 4, 24, 256, 3360, 53824, 1016064, 22095616, 543966720, 14955833344, 454227400704, 15103031627776, 545668238868480, 21286707282264064, 891735287528914944, 39926103010743156736

Offset: 0

Karol A. Penson, Apr 26 2010

			a(4) = 24: The 24 plane increasing trees on 4 vertices are
............................................................
.........1(x4 colors).......1(x4 colors).......1(x4 colors).
......../|\................/|\................/|\...........
......./.|.\............../.|.\............../.|.\..........
......2..3..4............2..4..3............3..2..4.........
............................................................
.........1(x4 colors).......1(x4 colors).......1(x4 colors).
......../|\................/|\................/|\...........
......./.|.\............../.|.\............../.|.\..........
......3..4..2............4..2..3............4..3..2.........
............................................................

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.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.

Cf. A000629, A000984, A124212, A124214, A229558.

max = 17; g[x_] := -(1/2)*Sqrt[2*Exp[-2*x] - 1] + 1/2; CoefficientList[ Series[ g[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Oct 05 2011 *)

x='x+O('x^66); concat ([0], Vec( serlaplace( serreverse( -1/2*log(1-2*x+2*x^2) ) ) ) ) \\ Joerg Arndt, Mar 01 2014

The e.g.f. A(x) satisfies the autonomous differential equation
A' = (1-2*A+2*A^2)/(1-2*A) with A(0) = 0. The compositional inverse of the e.g.f. is -1/2*log(1-2*x+2*x^2).
a(n) = (-1)^(n-1)*D^(n-1)(1) evaluated at x = 1, where D denotes the operator g(x) -> d/dx((x+1/x)*g(x)).
Applying [Bergeron et al., Theorem 1] to the result x = int {t = 0..A(x)} 1/phi(t), where phi(t) = (1-2*t+2*t^2)/(1-2*t) = 1+2*t^2+4*t^3+8*t^4+... leads to the following combinatorial interpretation for this sequence: a(n) gives the number of plane increasing trees on n vertices with no vertices of outdegree 1 and where each vertex of outdegree k >= 2 can be colored in 2^(k-1) ways. An example is given below. - Peter Bala, Sep 06 2011
a(n) ~ 2^(n-3/2)*n^(n-1)/(exp(n)*(log(2))^(n-1/2)). - Vaclav Kotesovec, Jun 28 2013
a(n+1) = 1/sqrt(2) * Sum_{k >= 0} (1/8)^k*binomial(2*k,k)*(2*k - 1)^n = 1/sqrt(2)*Sum_{k >= 0} (-1/2)^k*binomial(-1/2,k)*(2*k - 1)^n = Sum_{k = 0..n} Sum_{i = 0..k} (-1)^(k-i)/4^k* binomial(2*k,k)*binomial(k,i)*(2*i - 1)^n. Cf. A124212, A124214 and A229558. - Peter Bala, Aug 30 2016

A229559 E.g.f. A(x) satisfies: A'(x) = A(x) - A(x)^2 + A(x)^3.

Original entry on oeis.org

1, 1, 2, 8, 46, 340, 3080, 33020, 408760, 5737600, 90041480, 1562155760, 29688699040, 613378208080, 13687826477600, 328103392918400, 8407745335871200, 229363476365320000, 6636554484183747200, 203007064284501963200, 6545704564601880726400, 221887453210925123353600

Offset: 0

Paul D. Hanna, Dec 18 2013

nonn

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 46*x^4/4! + 340*x^5/5! +...
where A(x)^2 = 1 + 2*x + 6*x^2/2! + 28*x^3/3! + 180*x^4/4! + 1460*x^5/5! +...
also, A(x)^3 = 1 + 3*x + 12*x^2/2! + 66*x^3/3! + 474*x^4/4! + 4200*x^5/5! +...
and log(A(x)) = x + x^2/2! + 4*x^3/3! + 20*x^4/4! + 134*x^5/5! + 1120*x^6/6! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..150

Cf. A124212.

{a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A-A^2+A^3+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

E.g.f. A(x) satisfies: A(x) = exp(x + Integral A(x)^2 - A(x) dx).

a(n) ~ n^n * 3^(3*n/2+3/4) / (exp(n) * Pi^(n+1/2)). - Vaclav Kotesovec, Dec 19 2013

cp's OEIS Frontend

A124214 E.g.f.: exp(x) / (2 - exp(3*x))^(1/3).

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A186695 A Galton triangle: T(n,k) = (2k-1)*(T(n-1,k) + T(n-1,k-1)): a type B analog of the ordered Bell numbers A019538.

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A229558 E.g.f.: exp(x) / (2 - exp(4*x))^(1/4).

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A276371 Expansion of e.g.f. exp(x/2)/(2 - exp(2*x))^(1/4).

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A124213 Expansion of e.g.f.: exp(exp(x)/sqrt(2-exp(2*x))-1).

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A365777 Expansion of e.g.f. (exp(2*x) / (2 - exp(2*x)))^(3/4).

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A176785 Sequence with e.g.f. g(x) = -(1/2)*sqrt(2*exp(-2*x)-1) + 1/2.

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A229559 E.g.f. A(x) satisfies: A'(x) = A(x) - A(x)^2 + A(x)^3.

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A365777 Expansion of e.g.f. (exp(2x) / (2 - exp(2x)))^(3/4).

A176785 Sequence with e.g.f. g(x) = -(1/2)sqrt(2exp(-2*x)-1) + 1/2.