A007106 -id:A007106 - cp's OEIS frontend

A143599 E.g.f. satisfies: A(x) = exp( x*sqrt(A(x)/A(-x)) ).

1, 1, 3, 10, 53, 316, 2527, 22072, 239689, 2774800, 38284091, 553477024, 9284250109, 161180444608, 3187413648343, 64638167906176, 1473221217774353, 34190645940363520, 882759869810501491, 23079229227696318976

Offset: 0

Paul D. Hanna, Aug 27 2008

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +...
F(x) = sqrt(A(x)/A(-x)) = e.g.f. of A058014:
F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...
where F(x) = exp(x*(F(x) + 1/F(x))/2).
G(x) = [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601:
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
where G(x) = cosh(x*G(x)).
S(x) = [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106:
S(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...
where S(x) = sqrt(G(x)^2 - 1) and G(x) = e.g.f. of A143601.

Vaclav Kotesovec, Table of n, a(n) for n = 0..218

Cf. A058014, A143600, A143601, A007106.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x*sqrt(A/subst(A,x,-x))));n!*polcoeff(A,n)}

E.g.f.: A(x) = exp(x*exp(x*G(x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
E.g.f.: sqrt(A(x)/A(-x)) = F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
E.g.f.: [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601.
E.g.f.: [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106.
E.g.f.: A(x) = H(x/2)^2 where H(x) = exp(x*H(x)/H(-x)) = e.g.f. of A143600.
E.g.f. satisfies: A(x/cosh(x)) = exp(x*exp(x)/cosh(x)). [From Paul D. Hanna, Aug 29 2008]
E.g.f. satisfies: -x*sqrt(x^2/log(y)^2) = log((x^2*y)/log(y)^2), where y=A(x). - Vaclav Kotesovec, Feb 26 2014
a(n) ~ c * n! * d^n / n^(3/2), where d = 1.5088795615383199289... is the root of the equation sqrt(1+1/d^2) = 1 + LambertW((1+sqrt(1+1/d^2))/exp(1 + sqrt(1+1/d^2))), and c = 7.98255033020099890281693169... if n is even, and c = 7.852067808737280621088934789... if n is odd. - Vaclav Kotesovec, Feb 26 2014

A274699 Number of varied unrooted labeled trees on n nodes, i.e., trees with unique vertex degree except for the leaves.

Original entry on oeis.org

1, 1, 3, 4, 65, 126, 637, 21344, 57465, 330850, 2023901, 156312432, 502733101, 3464645380, 21505493115, 194182086016, 28828634104193, 108228186522378, 842870098110349, 5690185704481040, 49492420589959221, 521939648330427580, 151814675979620347079, 642254623640191155552, 5633344428589261601425, 40942919469553204283076, 370068971963735972601207

Offset: 1

Marko Riedel, Jul 02 2016

nonn

Math.Stackexchange.Com, Marko R. Riedel et al., Labeled trees with a certain condition
Marko Riedel, Maple code to compute sequence from the formula and by total enumeration

Cf. A007106.

a(n) = (n-2)! Sum_{k=1..n-2} C(n,k) k! [u^k][z^{n-2}] Product_{q=1..n-2} (1 + u z^q/q!).

A294198 Labeled trees on n nodes with at least one node of degree two.

Original entry on oeis.org

0, 0, 3, 12, 120, 1200, 16380, 255696, 4726008, 99107280, 2346042600, 61706210280, 1788467429892, 56618211155688, 1944581982268380, 72019509651227040, 2861473883255362800, 121414006972684901664, 5479661140682410928592, 262122044885503316203320

Offset: 1

Marko Riedel, Feb 11 2018

nonn

			When n=4 there are two types of trees: paths (24!/2  trees) or a star (4 trees) for a total of 4^(4-2) trees. Of these only the paths contain a vertex of degree 2 (2 in fact). When n=5 there are three types of trees: paths (5!/2 trees), a star (5 trees) or a fork (C(5,1) x C(4, 2) x 2) for a total of 5^(5-2) trees. Of these only the paths and the fork contain a vertex of degree 2.

Marko Riedel, Maple code for sequence including total enumeration and closed form.
Marko Riedel et al., How many trees have a vertex of degree two

Cf. A007106, A274699.

a[1] = 0; a[n_] := (n-2)! Sum[Binomial[n, q] (-1)^(q+1) (n-q)^(n-2-q) / (n-2-q)!, {q, 1, n-2}]; Array[a, 20] (* Jean-François Alcover, Feb 15 2018 *)

a(n) = (n-2)!*Sum_{q=1..n-2} C(n,q)*(-1)^(q+1)*(n-q)^(n-2-q)/(n-2-q)! where n >= 2.

A143598 E.g.f.: A(x) = exp(xsinh(xG(x))) where G(x) = cosh(x*G(x)) is the e.g.f. of A143601.

Original entry on oeis.org

1, 2, 28, 1176, 103440, 15726880, 3684098496, 1232799974784, 558670427013376, 329559835063067136, 245462725323910487040, 225319148634038399801344, 249936012383478860884217856, 329609037187846742271984869376

Offset: 0

Paul D. Hanna, Aug 27 2008

nonn

			E.g.f.: A(x) = 1 + 2*x^2/2! + 28*x^4/4! + 1176*x^6/6! + 103440*x^8/8! +...
A(x) = exp(x*F(x)) where F(x) = e.g.f. of A007106:
F(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...
A(x) = exp(x*sqrt(G(x)^2 - 1)) where G(x) = e.g.f. of A143601:
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
A(x) = sqrt(H(x)*H(-x)) where H(x) = e.g.f. of A143599:
H(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +...

Cf. A058014, A143600, A143601, A007106.

{a(n)=local(G=1+x*O(x^n));for(i=0,n,G=cosh(x*G));n!*polcoeff(exp(x*sqrt(G^2-1)),n)}

E.g.f.: A(x) = exp(x*F(x)) where F(x) is the e.g.f. of A007106.
E.g.f.: A(x) = sqrt(H(x)*H(-x)) where H(x) = exp(x*sqrt(H(x)/H(-x))) is the e.g.f. of A143599.
E.g.f. satisfies: A(x/cosh(x)) = exp(x*tanh(x)). [From Paul D. Hanna, Aug 29 2008]

cp's OEIS Frontend

A143599 E.g.f. satisfies: A(x) = exp( x*sqrt(A(x)/A(-x)) ).

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A274699 Number of varied unrooted labeled trees on n nodes, i.e., trees with unique vertex degree except for the leaves.

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Author

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Formula

A294198 Labeled trees on n nodes with at least one node of degree two.

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Mathematica

Formula

A143598 E.g.f.: A(x) = exp(xsinh(xG(x))) where G(x) = cosh(x*G(x)) is the e.g.f. of A143601.

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cp's OEIS Frontend

A143599 E.g.f. satisfies: A(x) = exp( x*sqrt(A(x)/A(-x)) ).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A274699 Number of varied unrooted labeled trees on n nodes, i.e., trees with unique vertex degree except for the leaves.

Views

Author

Keywords

Links

Crossrefs

Formula

A294198 Labeled trees on n nodes with at least one node of degree two.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A143598 E.g.f.: A(x) = exp(x*sinh(x*G(x))) where G(x) = cosh(x*G(x)) is the e.g.f. of A143601.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A143598 E.g.f.: A(x) = exp(xsinh(xG(x))) where G(x) = cosh(x*G(x)) is the e.g.f. of A143601.