A006351 -id:A006351 - cp's OEIS frontend

A058387 Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.

0, 1, 1, 2, 4, 8, 18, 40, 94, 224, 548, 1356, 3418, 8692, 22352, 57932, 151312, 397628, 1050992, 2791516, 7447972, 19950628, 53635310, 144664640, 391358274, 1061628772, 2887113478, 7869761108, 21497678430, 58841838912, 161356288874

Offset: 0

N. J. A. Sloane, Dec 20 2000

This is a series-parallel network: o-o; all other series-parallel networks are obtained by connecting two series-parallel networks in series or in parallel. See A000084 for examples.

Order is not considered significant in series configurations. - Andrew Howroyd, Dec 22 2020

			From _Andrew Howroyd_, Dec 22 2020: (Start)
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element (an edge) is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (oo).
a(3) = 2: (ooo), (o|oo).
a(4) = 4: (oooo), (o(o|oo)), (o|ooo), (oo|oo).
a(5) = 8: (ooooo), (oo(o|oo)), (o(o|ooo)), (o(oo|oo)), (o|oooo), (o|o(o|oo)),  (oo|ooo), (o|oo|oo).
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Steven R. Finch, Series-parallel networks
Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
John W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence v_n).
Index entries for sequences mentioned in Moon (1987)

A000084 is the case that multiple edges are allowed.
A058381 is the case that edges are labeled.
A339290 is the case that order is significant in series configurations.
Cf. A058385, A058386, A000311, A000669, A006351.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(s=p=vector(n)); p[1]=1; for(n=2, n, s[n]=EulerT(p[1..n])[n]; p[n]=vecsum(EulerT(s[1..n])[n-1..n])-s[n]); concat([0], p+s)} \\ Andrew Howroyd, Dec 22 2020

a(n) = A058385(n) + A058386(n).

A234289 E.g.f. satisfies: A(x) = 1 + A(x)^2 * Integral 1/A(x) dx.

Original entry on oeis.org

1, 1, 3, 17, 147, 1729, 25827, 468593, 10012083, 246287521, 6856204803, 213102768977, 7315460977107, 274894137157249, 11223280473993507, 494715928976218673, 23416019742035332083, 1184519963466363339361, 63774753426394808946243, 3641219528568659379843857

Offset: 0

Paul D. Hanna, Dec 22 2013

nonn

Compare to: G(x) = 1 + G(x)^2 * Integral 1/G(x)^2 dx, where G(x) is the e.g.f. of A006351, the number of series-parallel networks with n labeled edges.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 147*x^4/4! + 1729*x^5/5! +...
where A(x)^2 = 1 + 2*x + 8*x^2/2! + 52*x^3/3! + 484*x^4/4! + 5948*x^5/5! +...
Integral 1/A(x) dx = x - x^2/2! - x^3/3! - 5*x^4/4! - 41*x^5/5! - 469*x^6/6! +...
Further,
Series_Reversion(Integral 1/A(x) dx) = x + x^2/2 + 2*x^3/3 + 5*x^4/4 + 14*x^5/5 + 42*x^6/6 + 132*x^7/7 +...+ A000108(n-1)*x^n/n +...
where A000108(n) = binomial(2*n,n)/(n+1).

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A000108, A234290.

seq(n! * coeff(series(-2/LambertW(-1,-2*exp(x-2)), x, n+1), x, n), n = 0..10) # Vaclav Kotesovec, Dec 27 2013

CoefficientList[1 + InverseSeries[Series[2*x/(1+x) - Log[1+x], {x, 0, 20}], x],x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 27 2013 *)

{a(n)=local(A=1+x);for(i=1,n,A=1+A^2*intformal(1/(A+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

/* Explicit formula using Catalan function C(x) = 1 + x*C(x)^2: */
{a(n)=local(C=(1-sqrt(1-4*x+x^2*O(x^n)))/(2*x),A=1); A=1/deriv(serreverse(intformal(C))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* Explicit formula: 1 + Series_Reversion(2*x/(1+x) - log(1+x)): */
{a(n)=local(A=1,X=x+x^2*O(x^n)); A=1+serreverse(2*X/(1+X)-log(1+X)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

E.g.f.: 1 / ( d/dx Series_Reversion( Integral C(x) dx ) ), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x), is the Catalan function of A000108.
E.g.f.: 1 + Series_Reversion( 2*x/(1+x) - log(1+x) ).
E.g.f.: -2/LambertW(-1,-2*exp(x-2)). - Vaclav Kotesovec, Dec 27 2013
E.g.f.: A(x) = C( Integral 1/A(x) dx ), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x), is the Catalan function of A000108. - Paul D. Hanna, May 23 2019
a(n) ~ 2 * n^(n-1) / (exp(n) * (1-log(2))^(n-1/2)). - Vaclav Kotesovec, Dec 27 2013

A058478 Total number of interior nodes in all essentially series series-parallel networks with n labeled edges, multiple edges allowed.

Original entry on oeis.org

0, 0, 1, 5, 46, 534, 7596, 127756, 2479856, 54560512, 1341716960, 36468949824, 1085680795200, 35131589529152, 1227777836217856, 46086892351150592, 1849266301464495616, 78990342571085637120, 3578513340735623076864

Offset: 0

N. J. A. Sloane, Dec 20 2000

nonn

J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the e.g.f. I_S(x)).

Index entries for sequences mentioned in Moon (1987)

A058480 = A058478 + A058479.

E.g.f. = (xP'-P)/(1+U), where P = e.g.f. for A000311 and U that for A006351.

A058479 Total number of interior nodes in all essentially parallel series-parallel networks with n labeled edges, multiple edges allowed.

Original entry on oeis.org

0, 0, 0, 3, 32, 410, 6164, 107492, 2140368, 47990784, 1197523456, 32930028736, 989647215424, 32276598717376, 1135501305508608, 42865272243657216, 1728443263014370304, 74145986811618564608, 3371731055491925101568

Offset: 0

N. J. A. Sloane, Dec 20 2000

nonn

J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the e.g.f. I_P(x)).

Index entries for sequences mentioned in Moon (1987)

A058480 = A058478 + A058479.

E.g.f. = product of e.g.f.'s for A006351 and A058478.

A234294 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^4 dx.

Original entry on oeis.org

1, 1, 4, 40, 664, 15424, 460576, 16808320, 724904896, 36072438016, 2034328297984, 128223244372480, 8932539799788544, 681536817951791104, 56521548341146402816, 5062454448656689500160, 487013865350356256137216, 50082306316236214342844416, 5482502331779770770018893824

Offset: 0

Paul D. Hanna, Dec 24 2013

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 40*x^3/3! + 664*x^4/4! + 15424*x^5/5! +...
where A(4*log(1+x) - 3*x) = 1+x.
Related series:
A(x)^4 = 1 + 4*x + 28*x^2/2! + 328*x^3/3! + 5752*x^4/4! + 137056*x^5/5! +...
1/A(x)^4 = 1 - 4*x + 4*x^2/2! - 40*x^3/3! - 536*x^4/4! - 13216*x^5/5! +...
(d/dx Series_Reversion(Integral 1/A(x)^4 dx))^(1/4) begins:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 +...+ A002293(n)*x^n +...
where G(x) = 1 + x*G(x)^4.

Cf. A006351, A058562, A234295.

CoefficientList[1 + InverseSeries[Series[4*Log[1+x]-3*x, {x, 0, 20}], x],x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 26 2013 *)

{a(n)=local(A=1+x); for(i=1, n, A=1+A^4*intformal(1/(A^4+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

{a(n)=local(A=1, X=x+x^2*O(x^n)); A=1+serreverse(4*log(1+X) - 3*X); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* O.g.f. continued fraction: */
{a(n)=local(CF=1+x*O(x)); for(k=0, n, CF=1-(n-k+1)*x-3*(n-k+1)*x/CF); polcoeff(1+x/CF, n, x)}
for(n=0, 25, print1(a(n), ", "))

E.g.f.: 1 + Series_Reversion( 4*log(1+x) - 3*x ).
E.g.f.: -4/3*LambertW(-3/4*exp((x-3)/4)).
E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x)^4 dx ) )^(1/4), where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
E.g.f.: 1 / sqrt( d/dx Series_Reversion( Integral (1+2*x*C(2*x))^2/(1+x)^2 dx ) ), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x), is the Catalan function of A000108.
O.g.f.: 1 + x/(1-x - 3*x/(1-2*x - 3*2*x/(1-3*x - 3*3*x/(1-4*x - 3*4*x/(1-...))))), a continued fraction.
a(n) ~ 2 * n^(n-1) / (3 * exp(n) * (8*log(2)-4*log(3)-1)^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013

A058561 A 3-way generalization of series-parallel networks with n labeled edges.

Original entry on oeis.org

0, 1, 3, 12, 78, 708, 8256, 117624, 1980096, 38456736, 846413472, 20819693568, 565998548352, 16852047061632, 545372109629184, 19061178127458816, 715541912895773184, 28713061759257037824, 1226513716981332031488

Offset: 0

N. J. A. Sloane, Dec 26 2000

nonn

Cf. A058534, A000311, A006351.

spec := [ N, {N=Union(Z,S,P,Q), S=Set(Union(Z,P),card>=2), P=Set(Union(Z,Q),card>=2), Q=Set(Union(Z,S),card>=2)}, labeled ]; [seq(combstruct[count](spec,size=n), n=0..40)]; # N = A058561, S=A000311

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

Vladimir Kruchinin, May 04 2011

For solving the differential equation A'(x)=G(A(x)), where G(0)!=0,
a(n) = 1/n!*sum(pi(i) in P(2*n-1,n), T(n,i)*prod(j=1..n, g(k_j-1))),
where pi(i) is the partition of 2*n-1 into n parts in lexicographic order P(2*n-1,n).
G(x) = g(0)+g(1)*x+g(2)*x^2+...
Examples
A003422  A'(x)=A(x)+1/(1-x)
A000108  A'(x)=1/(1-2*A(x)),
A001147  A'(x)=1/(1-A(x))
A007489  A'(x)=A(x)+x/(1-x)^2+1.
A006351  B'(x)=(1+B(x))/(1-B(x))
A029768  A'(x)=log(1/(1-A(x)))+1.
A001662  B'(x)=1/(1+B(x))
A180254  A'(x)=(1-sqrt(1-4*A(x)))/2
Compare with A145271. There (j')^k = [(d/dx)^j g(x)]^k evaluated at x=0 gives formulas expressed in terms of the coefficients of the Taylor series g(x). If, instead, we express the formulas in terms of the coefficients of the power series of g(x), we obtain a row reversed array for A190015 since the partitions there are in reverse order to the ones here. Simply exchange (j!)^k * (j")^k for (j')^k, where (j")^k = [(d/dx)^j g(x) / j!]^k, to transform from one array to the other. E.g., R^3 g(x) = 1 (0')^1 (1')^3 + 4 (0')^2 (1')^1 (2')^1 + 1 (0')^3 (3')^1 = 1 (O")^1 (1")^3 + 4 (0")^2 (1")^1 2*(2")^1 + 1 (0")^1 3!*(3")^1 = 1 (O")^1 (1")^3 + 8 (0")^2 (1")^1 (2")^1 + 6 (0")^1 (3")^1, the fourth partition polynomial here. - Tom Copeland, Oct 17 2014

			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 */

A234295 E.g.f. satisfies: A(x) = 1 + A(x)^5 * Integral 1/A(x)^5 dx.

Original entry on oeis.org

1, 1, 5, 65, 1405, 42505, 1653125, 78578225, 4414067725, 286099718425, 21015972365525, 1725374840578625, 156560122048892125, 15559151967183795625, 1680744724811088153125, 196083244062052339084625, 24570430118524659881918125, 3291153805391398126661325625

Offset: 0

Paul D. Hanna, Dec 25 2013

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 65*x^3/3! + 1405*x^4/4! + 42505*x^5/5! +...
where A(5*log(1+x) - 4*x) = 1+x.
Related series:
A(x)^5 = 1 + 5*x + 45*x^2/2! + 685*x^3/3! + 15645*x^4/4! + 485645*x^5/5! +...
1/A(x)^5 = 1 - 5*x + 5*x^2/2! - 85*x^3/3! - 1595*x^4/4! - 50645*x^5/5! +...
(d/dx Series_Reversion(Integral 1/A(x)^5 dx))^(1/5) begins:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 +...+ A002294(n)*x^n +...
where G(x) = 1 + x*G(x)^5.

Cf. A006351, A058562, A234294.

CoefficientList[1 + InverseSeries[Series[5*Log[1+x]-4*x, {x, 0, 20}], x],x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 26 2013 *)

{a(n)=local(A=1+x); for(i=1, n, A=1+A^5*intformal(1/(A^5+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

{a(n)=local(A=1,X=x+x^2*O(x^n)); A=1+serreverse(5*log(1+X) - 4*X); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* O.g.f. continued fraction: */
{a(n)=local(CF=1+x*O(x)); for(k=0, n, CF=1-(n-k+1)*x-4*(n-k+1)*x/CF); polcoeff(1+x/CF, n, x)}
for(n=0, 25, print1(a(n), ", "))

E.g.f.: 1 + Series_Reversion( 5*log(1+x) - 4*x ).
E.g.f.: -5/4*LambertW(-4/5*exp((x-4)/5)).
E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x)^5 dx ) )^(1/5), where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.
O.g.f.: 1 + x/(1-x - 4*x/(1-2*x - 4*2*x/(1-3*x - 4*3*x/(1-4*x - 4*4*x/(1-...))))), a continued fraction.
a(n) ~ sqrt(5) * n^(n-1) / (4*exp(n)*(5*log(5)-10*log(2)-1)^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013

A278458 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 2, 2, 9, 15, 8, 64, 156, 144, 52, 625, 2050, 2675, 1730, 472, 7776, 32430, 55000, 50310, 25108, 5504, 117649, 599319, 1258775, 1484245, 1052184, 428036, 78416, 2097152, 12669496, 31902416, 46103680, 42064736, 24421096, 8389552, 1320064, 43046721, 301574340, 888996066, 1524644856, 1698413409, 1269814980, 625219644, 185935104, 25637824

Offset: 1

Gheorghe Coserea, Jan 15 2017

			A(x;t) = x + (2*t+2)*x^2/2! + (9*t^2+15*t+8)*x^3/3! + (64*t^3+156*t^2+144*t+52)*x^4/4! + ...
Triangle starts:
n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]
[1]  1;
[2]  2,       2;
[3]  9,       15,      8;
[4]  64,      156,     144,     52;
[5]  625,     2050,    2675,    1730,    472;
[6]  7776,    32430,   55000,   50310,   25108,   5504;
[7]  117649,  599319,  1258775, 1484245, 1052184, 428036,  78416;
[8]  ...

Gheorghe Coserea, Rows n = 1..101, flattened.
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

Column k=1 give A000169

m = 10;
(Reverse[CoefficientList[#, t]]& /@ CoefficientList[InverseSeries[Log[x + Exp[t Log[1+x]]] - (t-1) Log[1+x] - x + O[x]^m], x]) Range[0, m-1]! // Rest // Flatten (* Jean-François Alcover, Sep 28 2019 *)

N=10; x = 'x + O('x^N); t='t;
concat(apply(p->Vec(p), Vec(serlaplace(serreverse(log(x + exp(t*log(1+x))) - (t-1)*log(1+x) - x)))))

y(x;t) = Sum {n>=1} P_n(t)*x^n/n! satisfies x = log(y + exp(t*log(1+y))) - (t-1)*log(1+y) - y.

A006351(n) = P_n(0), A005172(n) = P_n(1), A231691(n) = P_n(2).

A302285 Expansion of 1/(1 - x - x/(1 - 2x - x/(1 - 3x - x/(1 - 4x - x/(1 - 5x - x/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 2, 7, 33, 185, 1170, 8121, 60846, 486753, 4125852, 36846557, 345205559, 3381126995, 34524194712, 366635359887, 4041180951473, 46149726728969, 545161967955668, 6652026230285141, 83730953689450825, 1085924693069106823, 14494802798426546660, 198918641942013097723

Offset: 0

Ilya Gutkovskiy, Apr 04 2018

nonn

a(n) is the number of paths from (0,0) to (2n,0) on or above the x-axis with steps U=(1,1), D=(1,-1), and L=(2,0), where the level steps L at height k have k+1 colors for all k>=0. - Alexander Burstein, Apr 10 2025

			G.f. A(x) = 1 + 2*x + 7*x^2 + 33*x^3 + 185*x^4 + 1170*x^5 + 8121*x^6 + 60846*x^7 + 486753*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..549
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.

Cf. A001515, A006318, A006351, A006789, A258173, A295289.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, b(x-1, y-1)+b(x-1, y+1)+b(x-2, y)*(y+1)))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2025

nmax = 22; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x, 1 - (k + 1) x, {k, 1, nmax}]), {x, 0, nmax}], x]

cp's OEIS Frontend

A058387 Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.

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A234289 E.g.f. satisfies: A(x) = 1 + A(x)^2 * Integral 1/A(x) dx.

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Maple

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PARI

PARI

PARI

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A058478 Total number of interior nodes in all essentially series series-parallel networks with n labeled edges, multiple edges allowed.

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Formula

A058479 Total number of interior nodes in all essentially parallel series-parallel networks with n labeled edges, multiple edges allowed.

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A234294 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^4 dx.

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Mathematica

PARI

PARI

PARI

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A058561 A 3-way generalization of series-parallel networks with n labeled edges.

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Maple

A190015 Triangle T(n,k) for solving differential equation A'(x)=G(A(x)), G(0)!=0.

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Maxima

Maxima

PARI

A234295 E.g.f. satisfies: A(x) = 1 + A(x)^5 * Integral 1/A(x)^5 dx.

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Mathematica

A302285 Expansion of 1/(1 - x - x/(1 - 2x - x/(1 - 3x - x/(1 - 4x - x/(1 - 5x - x/(1 - ...)))))), a continued fraction.