A007106 -id:A007106 - cp's OEIS frontend

A058014 Number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.

1, 1, 1, 4, 13, 96, 541, 5888, 47545, 686080, 7231801, 130179072, 1695106117, 36590059520, 567547087381, 14290429935616, 257320926233329, 7405376630685696, 151856004814953841, 4917457306800619520, 113144789723082206461, 4071967909087792857088

Offset: 0

Alex Postnikov (apost(AT)math.mit.edu), Nov 13 2000

			E.g.f.: A(x) = 1 + x + x^2/2! + 4x^3/3! + 13x^4/4! + 96x^5/5! +...

Seiichi Manyama, Table of n, a(n) for n = 0..422
Alexander Postnikov, Papers.
A. Postnikov and R. P. Stanley, Deformations of Coxeter hyperplane arrangements, J. Combin. Theory, Ser. A, 91 (2000), 544-597. (Section 10.2.)

Cf. bisections: A007106, A143601.

Cf. A138764 (variant).

a := n -> 2^(-n)*add(binomial(n,k)*(n+1-2*k)^(n-1), k=0..n);

a[n_] := Sum[((n-2k+1)^(n-1)*n!) / (k!*(n-k)!), {k, 0, n}] / 2^n; a[1] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 14 2011, after Maple *)

{a(n)=local(A=1+x);for(i=0,n,A=exp(x*(A+1/(A +x*O(x^n)))/2));n!*polcoeff(A,n)} \\ Paul D. Hanna, Mar 29 2008

{a(n) = sum(k=0, n, binomial(n, k)*(n+1-2*k)^(n-1))/2^n} \\ Seiichi Manyama, Sep 27 2020

a(n) = (1/2^n) * Sum_{k=0..n} binomial(n,k) * (n + 1 - 2*k)^(n-1).
From Paul D. Hanna, Mar 29 2008: (Start)
E.g.f. satisfies A(x) = exp( x*[A(x) + 1/A(x)]/2 ).
E.g.f. A(x) equals the inverse function of 2*x*log(x)/(1 + x^2).
Let r = radius of convergence of A(x), then r = 0.6627434193491815809747420971092529070562335491150224... and A(r) = 3.31905014223729720342271370055697247448941708369151595... where A(r) and r satisfy A(r) = exp( (A(r)^2 + 1)/(A(r)^2 - 1) ) and r = 2*A(r)/(A(r)^2 - 1). (End)
E.g.f. A(x)=exp(B(x)), B(x) satisfies B(x)=x*cosh(B(x)). [Vladimir Kruchinin, Apr 19 2011]
a(n) ~ (1-(-1)^n*s^2)/s * n^(n-1) * ((1-s^2)/(2*s))^n / exp(n), where s = 0.3012910191606573456... is the root of the equation (1+s^2) = (s^2-1)*log(s), r = 2*s/(1-s^2). - Vaclav Kotesovec, Jan 08 2014
E.g.f. satisfies A(-x) = 1/A(x). - Alexander Burstein, Oct 26 2021
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (2*k+1) * binomial(n-1,2*k) * a(2*k) * a(n-1-2*k). - Seiichi Manyama, Jul 05 2025

Updated URL and author's e-mail address - R. J. Mathar, May 23 2010

A143601 Number of labeled odd-degree trees with 2n+1 nodes.

Original entry on oeis.org

1, 1, 13, 541, 47545, 7231801, 1695106117, 567547087381, 257320926233329, 151856004814953841, 113144789723082206461, 103890621918675777804301, 115270544419577901796226473, 152049571406030636219959644841

Offset: 0

Paul D. Hanna, Aug 26 2008, May 27 2009

nonn

			E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! + ...
The e.g.f. of A007106 (a bisection of A058014) is given by:
sqrt(A(x)^2 - 1) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! + ...
The e.g.f. of A058014 is given by:
F(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! + ...
where A(x) = [F(x) + F(-x)]/2 and exp(x*A(x)) = F(x).
The e.g.f. of A143600 is given by:
G(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! + ...
where A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2.

Seiichi Manyama, Table of n, a(n) for n = 0..211

Cf. A058014, A143600, A007106, A309204.

Cf. A370430, A370432.

Table[(2*n)!*CoefficientList[1/x*InverseSeries[Series[x/Cosh[x],{x,0,41}],x],x][[2*n+1]],{n,0,20}] (* Vaclav Kotesovec, Jan 10 2014 *)

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=cosh(x*A));n!*polcoeff(A,n)}

{a(n)=(2*n)!*polcoeff(cosh(x+x*O(x^(2*n)))^(2*n+1)/(2*n+1),2*n)} \\ Paul D. Hanna, Aug 29 2008

{a(n) = sum(k=0,n, binomial(2*n+1,k) * (2*n+1-2*k)^(2*n) / ((2*n+1) * 2^(2*n)) )}
for(n=0,30, print1(a(n),", ")) \\ Paul D. Hanna, Feb 19 2024

E.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas.
(1) A(x) = cosh(x*A(x)).
(2) A(x) = (1/x)*Series_Reversion( x/cosh(x) ).
(3) sqrt(A(x)^2 - 1) = e.g.f. of A007106.
(4) exp(x*A(x)) = A(x) + sqrt(A(x)^2-1) = e.g.f. of A058014.
(5) A(x) = [F(x) + F(-x)]/2 where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
(6) A(2*x) = [G(x)/G(-x) + G(-x)/G(x)]/2 where G(x) = exp(x*G(x)/G(-x)) = e.g.f. of A143600.
From Paul D. Hanna, Aug 29 2008: (Start)
(7) A(x/cosh(x)) = cosh(x).
(8) a(n) = (2n)!*[x^(2n)] cosh(x)^(2n+1)/(2n+1). (End)
(9) a(n) = Sum_{k=0..n} binomial(2*n+1,k) * (2*n+1 - 2*k)^(2*n) / ((2*n+1) * 2^(2*n)). [See formula by Christophe Vignat in A309204.] - Paul D. Hanna, Feb 19 2024
a(n) ~ 2^(2*n) * n^(2*n-1) * (s^2-1)^(n+1/2) / exp(2*n), where s = 1.810170580698977274512829... is the root of the equation sqrt(s^2-1) * log(s + sqrt(s^2-1)) = s. - Vaclav Kotesovec, Jan 10 2014
Radius of convergence r = 0.66274341934918158097474... = 1/sqrt(s^2-1) and A(r) = s (given above) satisfy r = 1/sinh(r*A(r)) and A(r) = cosh(r*A(r)). - Paul D. Hanna, Mar 04 2024

Edited by Paul D. Hanna, May 27 2009

A370431 Expansion of e.g.f. S(x,k) satisfying S(x,k) = sinh( xcosh(kx*sqrt(1 + S(x,k)^2)) ), as a triangle read by rows.

Original entry on oeis.org

1, 1, 3, 1, 90, 5, 1, 2205, 3675, 7, 1, 46116, 532350, 107604, 9, 1, 812295, 52887450, 74042430, 2436885, 11, 1, 12666654, 4257556875, 24609789204, 7663602375, 46444398, 13, 1, 181355265, 292686719325, 5841878527485, 7510986678195, 643910782515, 785872815, 15, 1, 2439315720, 17658076954700, 1124109212938712, 4451226370197750, 1766457334617976, 45911000082220, 12196578600, 17

Offset: 0

Paul D. Hanna, Feb 19 2024

The row sums equal A007106, the number of labeled odd degree trees with 2n nodes.
Unsigned version of triangle A370331.
A row reversal of triangle A370433.

			E.g.f.: S(x,k) = x + (1 + 3*k^2)*x^3/3! + (1 + 90*k^2 + 5*k^4)*x^5/5! + (1 + 2205*k^2 + 3675*k^4 + 7*k^6)*x^7/7! + (1 + 46116*k^2 + 532350*k^4 + 107604*k^6 + 9*k^8)*x^9/9! + (1 + 812295*k^2 + 52887450*k^4 + 74042430*k^6 + 2436885*k^8 + 11*k^10)*x^11/11! + (1 + 12666654*k^2 + 4257556875*k^4 + 24609789204*k^6 + 7663602375*k^8 + 46444398*k^10 + 13*k^12)*x^13/13! + ...
where S(x,k) = sinh( x*cosh(k*x*sqrt(1 + S(x,k)^2)) ).
This triangle of coefficients a(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in S(x,k) begins
 1;
 1, 3;
 1, 90, 5;
 1, 2205, 3675, 7;
 1, 46116, 532350, 107604, 9;
 1, 812295, 52887450, 74042430, 2436885, 11;
 1, 12666654, 4257556875, 24609789204, 7663602375, 46444398, 13;
 1, 181355265, 292686719325, 5841878527485, 7510986678195, 643910782515, 785872815, 15;
 1, 2439315720, 17658076954700, 1124109212938712, 4451226370197750, 1766457334617976, 45911000082220, 12196578600, 17;
 1, 31284206667, 957418671788100, 185331614609361948, 1972848836100689118, 2411259688567508922, 344187284274529332, 2872256015364300, 177277171113, 19; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1325

Cf. A370430 (C), A370432 (D), A370433 (T), A007106 (row sums).

Cf. A370331.

{a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n+1)));
for(i=1, 2*n+1,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))););
(2*n+1)! *polcoeff(polcoeff(S, 2*n+1, x), 2*j, k)}
for(n=0, 10, for(k=0, n, print1( a(n, k), ", ")); print(""))

E.g.f.: S(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n+1)*k^(2*j)/(2*n+1)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + k*T) = exp(k*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - k^2*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(k*x*C).
(3.d) T = (1/k) * sinh(k*x*C).
(4.a) C = cosh( x*cosh(k*x*C) ).
(4.b) S = sinh( x*cosh(k*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( k*x*cosh(x*D) ).
(4.d) T = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T^2)) ).
(5.a) (C*D + k*S*T) = cosh(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sinh(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 + Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 + k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = S*S'.
(8.b) D*D' = k^2*T*T'.
(9.a) C' = S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = S * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = k^2 * T * (C + x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
Logarithms.
(12.a) D = log(C + sqrt(C^2 - 1)) / x.
(12.b) C = log(D + sqrt(D^2 - 1)) / (k*x).
(12.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (k*x).
(12.d) S = sqrt(log(k*T + sqrt(1 + k^2*T^2))^2 - k^2*x^2) / (k*x).

A370433 Expansion of e.g.f. T(x,k) satisfying T(x,k) = (1/k) * sinh( kxcosh(xsqrt(1 + k^2T(x,k)^2)) ), as a triangle read by rows.

Original entry on oeis.org

1, 3, 1, 5, 90, 1, 7, 3675, 2205, 1, 9, 107604, 532350, 46116, 1, 11, 2436885, 74042430, 52887450, 812295, 1, 13, 46444398, 7663602375, 24609789204, 4257556875, 12666654, 1, 15, 785872815, 643910782515, 7510986678195, 5841878527485, 292686719325, 181355265, 1, 17, 12196578600, 45911000082220, 1766457334617976, 4451226370197750, 1124109212938712, 17658076954700, 2439315720, 1

Offset: 0

Paul D. Hanna, Feb 19 2024

The row sums equal A007106, the number of labeled odd degree trees with 2n nodes.
Unsigned version of triangle A370333.
A row reversal of triangle A370431.

			E.g.f.: T(x,k) = x + (3 + k^2)*x^3/3! + (5 + 90*k^2 + k^4)*x^5/5! + (7 + 3675*k^2 + 2205*k^4 + k^6)*x^7/7! + (9 + 107604*k^2 + 532350*k^4 + 46116*k^6 + k^8)*x^9/9! + (11 + 2436885*k^2 + 74042430*k^4 + 52887450*k^6 + 812295*k^8 + k^10)*x^11/11! + (13 + 46444398*k^2 + 7663602375*k^4 + 24609789204*k^6 + 4257556875*k^8 + 12666654*k^10 + k^12)*x^13/13! + ...
where T(x,k) = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T(x,k)^2)) ).
This triangle of coefficients a(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in T(x,k) begins
 1;
 3, 1;
 5, 90, 1;
 7, 3675, 2205, 1;
 9, 107604, 532350, 46116, 1;
 11, 2436885, 74042430, 52887450, 812295, 1;
 13, 46444398, 7663602375, 24609789204, 4257556875, 12666654, 1;
 15, 785872815, 643910782515, 7510986678195, 5841878527485, 292686719325, 181355265, 1;
 17, 12196578600, 45911000082220, 1766457334617976, 4451226370197750, 1124109212938712, 17658076954700, 2439315720, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1325

Cf. A370430 (C), A370431 (S), A370432 (D), A007106 (row sums).

Cf. A370333.

{a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n+1)));
for(i=1, 2*n+1,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))););
(2*n+1)! *polcoeff(polcoeff(T, 2*n+1, x), 2*j, k)}
for(n=0, 10, for(k=0, n, print1( a(n, k), ", ")); print(""))

E.g.f.: T(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n+1)*k^(2*j)/(2*n+1)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + k*T) = exp(k*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - k^2*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(k*x*C).
(3.d) T = (1/k) * sinh(k*x*C).
(4.a) C = cosh( x*cosh(k*x*C) ).
(4.b) S = sinh( x*cosh(k*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( k*x*cosh(x*D) ).
(4.d) T = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T^2)) ).
(5.a) (C*D + k*S*T) = cosh(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sinh(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 + Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 + k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = S*S'.
(8.b) D*D' = k^2*T*T'.
(9.a) C' = S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = S * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = k^2 * T * (C + x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
Logarithms.
(12.a) D = log(C + sqrt(C^2 - 1)) / x.
(12.b) C = log(D + sqrt(D^2 - 1)) / (k*x).
(12.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (k*x).
(12.d) S = sqrt(log(k*T + sqrt(1 + k^2*T^2))^2 - k^2*x^2) / (k*x).

A143600 E.g.f. satisfies: A(x) = exp(x*A(x)/A(-x)).

Original entry on oeis.org

1, 1, 5, 25, 249, 2561, 40573, 641817, 14110001, 302279617, 8530496181, 230851019609, 7964867290537, 260618470319169, 10635790073585069, 408342804482252761, 19246730825243728737, 848289638051491455617, 45356940470607637151845, 2257054105205570995111833

Offset: 0

Paul D. Hanna, Aug 26 2008

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! +...
A LambertW identity yields the series:
A(x) = 1 + x/A(-x) + 3^1*x^2/2!/A(-x)^2 + 4^2*x^3/3!/A(-x)^3 + 5^3*x^4/4!/A(-x)^4 + 6^4*x^5/5!/A(-x)^5 +...+ (n+1)^(n-1)*x^n/n!/A(-x)^n +...
RELATED EXPANSIONS.
A(x)/A(-x) = F(2x) where F(x) is the e.g.f. of A058014:
A(x)/A(-x) = 1 + 2*x + 4*x^2/2! + 32*x^3/3! + 208*x^4/4! + 3072*x^5/5! +...
F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...
which satisfies: F(x) = exp(x*(F(x) + 1/F(x))/2).
(A(x)/A(-x) + A(-x)/A(x))/2 = G(2x) where G(x) is the e.g.f. of A143601:
(A(x)/A(-x) + A(-x)/A(x))/2 = 1 + 4*x^2/2! + 208*x^4/4! + 34624*x^6/6! +...
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
which satisfies G(x) = cosh(x*G(x)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..175

Cf. A058014, A143601, A007106.

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

/* Formula Using a LambertW Identity: */
{a(n)=local(A=1);for(i=1,n,A=sum(k=0,n,(k+1)^(k-1)*x^k/k!/subst(A,x,-x)^k+x*O(x^n)));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 05 2012

E.g.f. A(x) satisfies:
(1) A(x) = exp(x*exp(2x*G(2x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
(2) [A(x)/A(-x) + A(-x)/A(x)]/2 = G(2x) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
(3) A(x)/A(-x) = exp(x*[A(x)/A(-x) + A(-x)/A(x)]) = F(2x) where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
(4) A(x) = Sum_{n>=0} (n+1)^(n-1) * x^n/n! / A(-x)^n.
(5) A(x)^m = Sum_{n>=0} m*(n+m)^(n-1) * x^n/n! / A(-x)^n.
(6) log(A(x)) = Sum_{n>=1} n^(n-1) * x^n/n! / A(-x)^n = x*A(x)/A(-x).
Formulas (4), (5), and (6) are due to LambertW identities. - Paul D. Hanna, Nov 05 2012
a(n) ~ c * n! / (n^(3/2) * r^n), where r = 0.33137170967459079... is the root of the equation sqrt(1+4*r^2) = log((1+sqrt(1+4*r^2))/(2*r)), and c = 1.35397895306096963692514418... if n is even, and c = 1.281887793570420328585518150... if n is odd. - Vaclav Kotesovec, Feb 25 2014

A370331 Expansion of e.g.f. S(x,k) satisfying S(x,k) = sin( xcos(kx*sqrt(1 - S(x,k)^2)) ), as a triangle read by rows.

Original entry on oeis.org

1, -1, -3, 1, 90, 5, -1, -2205, -3675, -7, 1, 46116, 532350, 107604, 9, -1, -812295, -52887450, -74042430, -2436885, -11, 1, 12666654, 4257556875, 24609789204, 7663602375, 46444398, 13, -1, -181355265, -292686719325, -5841878527485, -7510986678195, -643910782515, -785872815, -15

Offset: 0

Paul D. Hanna, Feb 19 2024

The unsigned row sums equal A007106.
Signed version of triangle A370431.
A row reversal of triangle A370333.

			E.g.f.: S(x,k) = x - (1 + 3*k^2)*x^3/3! + (1 + 90*k^2 + 5*k^4)*x^5/5! - (1 + 2205*k^2 + 3675*k^4 + 7*k^6)*x^7/7! + (1 + 46116*k^2 + 532350*k^4 + 107604*k^6 + 9*k^8)*x^9/9! - (1 + 812295*k^2 + 52887450*k^4 + 74042430*k^6 + 2436885*k^8 + 11*k^10)*x^11/11! + (1 + 12666654*k^2 + 4257556875*k^4 + 24609789204*k^6 + 7663602375*k^8 + 46444398*k^10 + 13*k^12)*x^13/13! + ...
where S(x,k) = sin( x*cos(k*x*sqrt(1 - S(x,k)^2)) ).
This triangle of coefficients a(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in S(x,k) begins
 1;
 -1, -3;
 1, 90, 5;
 -1, -2205, -3675, -7;
 1, 46116, 532350, 107604, 9;
 -1, -812295, -52887450, -74042430, -2436885, -11;
 1, 12666654, 4257556875, 24609789204, 7663602375, 46444398, 13;
 -1, -181355265, -292686719325, -5841878527485, -7510986678195, -643910782515, -785872815, -15;
 1, 2439315720, 17658076954700, 1124109212938712, 4451226370197750, 1766457334617976, 45911000082220, 12196578600, 17; ...

Cf. A370330 (C), A370332 (D), A370333 (T).

Cf. A370431.

{a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n+1)));
for(i=1, 2*n+1,
C = cos( x*cos(k*x*C +Ox) );
S = sin( x*cos(k*x*sqrt(1 - S^2 +Ox)) );
D = cos( k*x*cos(x*D +Ox));
T = (1/k)*sin( k*x*cos(x*sqrt(1 - k^2*T^2 +Ox))););
(2*n+1)! *polcoeff(polcoeff(S, 2*n+1, x), 2*j, k)}
for(n=0, 10, for(k=0, n, print1( a(n, k), ", ")); print(""))

E.g.f.: S(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n+1)*k^(2*j)/(2*n+1)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + i*S) = exp(i*x*D).
(1.b) (D + i*k*T) = exp(i*k*x*C).
(2.a) C^2 + S^2 = 1.
(2.b) D^2 + k^2*T^2 = 1.
Circular functions.
(3.a) C = cos(x*D).
(3.b) S = sin(x*D).
(3.c) D = cos(k*x*C).
(3.d) T = (1/k) * sin(k*x*C).
(4.a) C = cos( x*cos(k*x*C) ).
(4.b) S = sin( x*cos(k*x*sqrt(1 - S^2)) ).
(4.c) D = cos( k*x*cos(x*D) ).
(4.d) T = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T^2)) ).
(5.a) (C*D - k*S*T) = cos(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sin(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 - Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 - k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = -S*S'.
(8.b) D*D' = -k^2*T*T'.
(9.a) C' = -S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = -k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = -S * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = -k^2 * T * (C - x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).

A370333 Expansion of e.g.f. T(x,k) satisfying T(x,k) = (1/k) * sin( kxcos(xsqrt(1 - k^2T(x,k)^2)) ), as a triangle read by rows.

Original entry on oeis.org

1, -3, -1, 5, 90, 1, -7, -3675, -2205, -1, 9, 107604, 532350, 46116, 1, -11, -2436885, -74042430, -52887450, -812295, -1, 13, 46444398, 7663602375, 24609789204, 4257556875, 12666654, 1, -15, -785872815, -643910782515, -7510986678195, -5841878527485, -292686719325, -181355265, -1

Offset: 0

Paul D. Hanna, Feb 19 2024

Unsigned row sums equal A007106.
Signed version of triangle A370433.
A row reversal of triangle A370331.

			E.g.f.: T(x,k) = x - (3 + k^2)*x^3/3! + (5 + 90*k^2 + k^4)*x^5/5! - (7 + 3675*k^2 + 2205*k^4 + k^6)*x^7/7! + (9 + 107604*k^2 + 532350*k^4 + 46116*k^6 + k^8)*x^9/9! - (11 + 2436885*k^2 + 74042430*k^4 + 52887450*k^6 + 812295*k^8 + k^10)*x^11/11! + (13 + 46444398*k^2 + 7663602375*k^4 + 24609789204*k^6 + 4257556875*k^8 + 12666654*k^10 + k^12)*x^13/13! + ...
where T(x,k) = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T(x,k)^2)) ).
This triangle of coefficients a(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in T(x,k) begins
 1;
 -3, -1;
 5, 90, 1;
 -7, -3675, -2205, -1;
 9, 107604, 532350, 46116, 1;
 -11, -2436885, -74042430, -52887450, -812295, -1;
 13, 46444398, 7663602375, 24609789204, 4257556875, 12666654, 1;
 -15, -785872815, -643910782515, -7510986678195, -5841878527485, -292686719325, -181355265, -1;
 17, 12196578600, 45911000082220, 1766457334617976, 4451226370197750, 1124109212938712, 17658076954700, 2439315720, 1; ...

Cf. A370330 (C), A370331 (S), A370332 (D).

Cf. A370433.

{a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n+1)));
for(i=1, 2*n+1,
C = cos( x*cos(k*x*C +Ox) );
S = sin( x*cos(k*x*sqrt(1 - S^2 +Ox)) );
D = cos( k*x*cos(x*D +Ox));
T = (1/k)*sin( k*x*cos(x*sqrt(1 - k^2*T^2 +Ox))););
(2*n+1)! *polcoeff(polcoeff(T, 2*n+1, x), 2*j, k)}
for(n=0, 10, for(k=0, n, print1( a(n, k), ", ")); print(""))

E.g.f.: T(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n+1)*k^(2*j)/(2*n+1)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + i*S) = exp(i*x*D).
(1.b) (D + i*k*T) = exp(i*k*x*C).
(2.a) C^2 + S^2 = 1.
(2.b) D^2 + k^2*T^2 = 1.
Circular functions.
(3.a) C = cos(x*D).
(3.b) S = sin(x*D).
(3.c) D = cos(k*x*C).
(3.d) T = (1/k) * sin(k*x*C).
(4.a) C = cos( x*cos(k*x*C) ).
(4.b) S = sin( x*cos(k*x*sqrt(1 - S^2)) ).
(4.c) D = cos( k*x*cos(x*D) ).
(4.d) T = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T^2)) ).
(5.a) (C*D - k*S*T) = cos(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sin(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 - Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 - k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = -S*S'.
(8.b) D*D' = -k^2*T*T'.
(9.a) C' = -S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = -k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = -S * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = -k^2 * T * (C - x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).

A372812 Expansion of e.g.f. S(x) satisfying S(x) = sinh( xcosh( 2x*sqrt(1 + S(x)^2) ) ), where a(n) is the coefficient of x^(2n+1)/(2n+1)! in S(x) for n >= 0.

Original entry on oeis.org

1, 13, 441, 68069, 15591025, 6212017725, 3652639410473, 2963960104898581, 3208843075117716705, 4442917542274682028653, 7676236962804930027455641, 16182752346241750118582151237, 40883629770018829153233694565201, 121951983267795526035606825074967709

Offset: 0

Paul D. Hanna, May 16 2024

nonn

			E.g.f: S(x) = x + 13*x^3/3! + 441*x^5/5! + 68069*x^7/7! + 15591025*x^9/9! + 6212017725*x^11/11! + 3652639410473*x^13/13! + 2963960104898581*x^15/15! + ...
and S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ).
RELATED SERIES.
Related functions C(x), D(x), and T(x) are described below.
C(x) = 1 + x^2/2! + 49*x^4/4! + 3601*x^6/6! + 680737*x^8/8! + 218915041*x^10/10! + 105958624465*x^12/12! + 74506995584113*x^14/14! + ...
where C(x) = sqrt(1 + S(x)^2)
and C(x) = cosh( x*cosh(2*x*C(x)) ).
D(x) = 1 + 4*x^2/2! + 64*x^4/4! + 7264*x^6/6! + 1242112*x^8/8! + 396112384*x^10/10! + 195196856320*x^12/12! + 135610245824512*x^14/14! + ...
where D(x) = cosh( 2*x*sqrt(1 + S(x)^2) )
and D(x) = cosh( 2*x*cosh(x*D(x)) ).
T(x) = x + 7*x^3/3! + 381*x^5/5! + 50051*x^7/7! + 11899705*x^9/9! + 4787171775*x^11/11! + 2800735142453*x^13/13! + 2286983798222779*x^15/15! + ...
where T(x) = (1/2) * sinh( 2*x*sqrt(1 + S(x)^2) )
and T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ).
SPECIFIC VALUES.
S(1/3) = 0.438594611804336870818029761992727975330083659221250216...
S(1/4) = 0.288479916487512228329919975913022787931012140199922189...
S(1/5) = 0.218707961000324022488369693038572482223647706535551198...
S(1/6) = 0.177223127385698497600070746700827976044841583345600952...
S(1/10) = 0.102204811824008710495811173453365253815203645781101342...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A370431 (k = 2), A372811 (C(x)), A372813 (D(x)), A372814 (T(x)), A007106.

/* From S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ) */
{a(n) = my(S=x); for(i=0,n, S=truncate(S); S = sinh( x*cosh(2*x*sqrt(1 + S^2 + x*O(x^(2*i)) )) ));
(2*n+1)! * polcoeff(S, 2*n+1, x)}
for(n=0, 30, print1( a(n), ", "))

/* From A370431 at k = 2 */
{a(n, k = 2) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
(2*n+1)! * polcoeff(S, 2*n+1, x)}
for(n=0, 30, print1( a(n), ", "))

a(n) = Sum_{j=0..n} A370431(n,j) * 2^(2*j).
E.g.f.: S(x) = Sum_{n>=0} a(n) * x^(2*n+1)/(2*n+1)! along with related functions denoted by C = C(x), S = S(x), D = D(x), and T = T(x) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + 2*T) = exp(2*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - 4*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(2*x*C).
(3.d) T = (1/2) * sinh(2*x*C).
(4.a) C = cosh( x*cosh(2*x*C) ).
(4.b) S = sinh( x*cosh(2*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( 2*x*cosh(x*D) ).
(4.d) T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2)) ).
(5.a) (C*D + 2*S*T) = cosh(x*D + 2*x*C).
(5.b) (S*D + 2*C*T) = sinh(x*D + 2*x*C).
Integrals.
(6.a) C = 1 + Integral S*D + x*S*D' dx.
(6.b) S = Integral C*D + x*C*D' dx.
(6.c) D = 1 + 4 * Integral T*C + x*T*C' dx.
(6.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(7.a) C*C' = S*S'.
(7.b) D*D' = 4*T*T'.
(8.a) C' = S * (D + x*D').
(8.b) S' = C * (D + x*D').
(8.c) D' = 4 * T * (C + x*C').
(8.d) T' = D * (C + x*C').
(9.a) C' = S * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.b) S' = C * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.c) D' = 4 * T * (C + x*S*D) / (1 - 4*x^2*S*T).
(9.d) T' = D * (C + x*S*D) / (1 - 4*x^2*S*T).
(10.a) (C + x*C') = (C + x*S*D) / (1 - 4*x^2*S*T).
(10.b) (D + x*D') = (D + 4*x*T*C) / (1 - 4*x^2*S*T).
Logarithms.
(11.a) D = log(C + sqrt(C^2 - 1)) / x.
(11.b) C = log(D + sqrt(D^2 - 1)) / (2*x).
(11.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (2*x).
(11.d) S = sqrt(log(2*T + sqrt(1 + 4*T^2))^2 - 4*x^2) / (2*x).
The radius of convergence r of e.g.f. S(x) is r = 0.458693345589772637742719473602361341151810356245785213... where S(r) = 1.201251917668278563521948977625996579820943724944393208...

A372814 Expansion of e.g.f. T(x) satisfying T(x) = (1/2) * sinh( 2xcosh( xsqrt(1 + 4T(x)^2) ) ), where a(n) is the coefficient of x^(2n+1)/(2n+1)! in T(x) for n >= 0.

Original entry on oeis.org

1, 7, 381, 50051, 11899705, 4787171775, 2800735142453, 2286983798222779, 2476757127978318705, 3434360322639603491447, 5940446259665147492879341, 12533181362722474751715110643, 31687559294370295303515685200041, 94578054008984518849163257005668911

Offset: 0

Paul D. Hanna, May 16 2024

nonn

			E.g.f: T(x) = x + 7*x^3/3! + 381*x^5/5! + 50051*x^7/7! + 11899705*x^9/9! + 4787171775*x^11/11! + 2800735142453*x^13/13! + 2286983798222779*x^15/15! + ...
and T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ).
RELATED SERIES.
Related functions C(x), S(x), and D(x) are described below.
C(x) = 1 + x^2/2! + 49*x^4/4! + 3601*x^6/6! + 680737*x^8/8! + 218915041*x^10/10! + 105958624465*x^12/12! + 74506995584113*x^14/14! + ...
where C(x) = cosh( x*sqrt(1 + 4*T(x)^2) )
and C(x) = cosh( x*cosh(2*x*C(x)) ).
S(x) = x + 13*x^3/3! + 441*x^5/5! + 68069*x^7/7! + 15591025*x^9/9! + 6212017725*x^11/11! + 3652639410473*x^13/13! + 2963960104898581*x^15/15! + ...
where S(x) = sinh( x*sqrt(1 + 4*T(x)^2) )
and S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ).
D(x) = 1 + 4*x^2/2! + 64*x^4/4! + 7264*x^6/6! + 1242112*x^8/8! + 396112384*x^10/10! + 195196856320*x^12/12! + 135610245824512*x^14/14! + ...
where D(x) = sqrt(1 + 4*T(x)^2)
and D(x) = cosh( 2*x*cosh(x*D(x)) ).
SPECIFIC VALUES.
T(1/3) = 0.396995956737823895057073833881324565170496359298875629...
T(1/4) = 0.272098438758403062037693234896743179247292418834110582...
T(1/5) = 0.210496129492378206205507948355083031094691532837329579...
T(1/6) = 0.172515307079869202329210004732936197729646250794708377...
T(1/10) = 0.101199443780546489307264980334019941975165460819500909...

Paul D. Hanna, Table of n, a(n) for n = 0..299

Cf. A370433 (k = 2), A372811 (C(x)), A372812 (S(x)), A372813 (D(x)), A007106.

/* From T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ) */
{a(n) = my(T=x); for(i=0,n, T=truncate(T); T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2 + x*O(x^(2*i)) )) ));
(2*n+1)! * polcoeff(T, 2*n+1, x)}
for(n=0, 30, print1( a(n), ", "))

/* From A370433 at k = 2 */
{a(n, k = 2) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
(2*n+1)! * polcoeff(T, 2*n+1, x)}
for(n=0, 30, print1( a(n), ", "))

a(n) = Sum_{j=0..n} A370433(n,j) * 2^(2*j).
E.g.f.: T(x) = Sum_{n>=0} a(n) * x^(2*n+1)/(2*n+1)! along with related functions denoted by C = C(x), S = S(x), D = D(x), and T = T(x) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + 2*T) = exp(2*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - 4*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(2*x*C).
(3.d) T = (1/2) * sinh(2*x*C).
(4.a) C = cosh( x*cosh(2*x*C) ).
(4.b) S = sinh( x*cosh(2*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( 2*x*cosh(x*D) ).
(4.d) T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2)) ).
(5.a) (C*D + 2*S*T) = cosh(x*D + 2*x*C).
(5.b) (S*D + 2*C*T) = sinh(x*D + 2*x*C).
Integrals.
(6.a) C = 1 + Integral S*D + x*S*D' dx.
(6.b) S = Integral C*D + x*C*D' dx.
(6.c) D = 1 + 4 * Integral T*C + x*T*C' dx.
(6.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(7.a) C*C' = S*S'.
(7.b) D*D' = 4*T*T'.
(8.a) C' = S * (D + x*D').
(8.b) S' = C * (D + x*D').
(8.c) D' = 4 * T * (C + x*C').
(8.d) T' = D * (C + x*C').
(9.a) C' = S * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.b) S' = C * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.c) D' = 4 * T * (C + x*S*D) / (1 - 4*x^2*S*T).
(9.d) T' = D * (C + x*S*D) / (1 - 4*x^2*S*T).
(10.a) (C + x*C') = (C + x*S*D) / (1 - 4*x^2*S*T).
(10.b) (D + x*D') = (D + 4*x*T*C) / (1 - 4*x^2*S*T).
Logarithms.
(11.a) D = log(C + sqrt(C^2 - 1)) / x.
(11.b) C = log(D + sqrt(D^2 - 1)) / (2*x).
(11.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (2*x).
(11.d) S = sqrt(log(2*T + sqrt(1 + 4*T^2))^2 - 4*x^2) / (2*x).
The radius of convergence r of e.g.f. T(x) is r = 0.458693345589772637742719473602361341151810356245785213... where T(r) = 0.989147448863398861152401518907145018698758995598697027...

A060279 Number of labeled rooted trees with all 2n nodes of odd degree.

Original entry on oeis.org

2, 16, 576, 47104, 6860800, 1562148864, 512260833280, 228646878969856, 133296779352342528, 98349146136012390400, 89583293999931442855936, 98732413018143104723582976, 129497500112719525122855141376, 199333356644821012200519079297024

Offset: 1

Vladeta Jovovic, Mar 28 2001

There are no such trees with an odd number of nodes.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

G. C. Greubel, Table of n, a(n) for n = 1..210

Cf. A007106.

A060279:= func< n | n eq 1 select 2 else n*(&+[Binomial(2*n,k)*(n-k)^(2*n-2) : k in [0..n-1]]) >;
[A060279(n): n in [1..30]]; // G. C. Greubel, Nov 05 2024

a:= j-> (n-> (n/2^n)*add(binomial(n, k)*(n-2*k)^(n-2), k=0..n))(2*j):
seq(a(n), n=1..15);  # Alois P. Heinz, Sep 27 2020

Flatten[{2,Table[n/2^n*Sum[Binomial[n,k]*(n-2*k)^(n-2),{k,0,n}],{n,4,30,2}]}] (* Vaclav Kotesovec, Jan 23 2014 *)
A060279[n_]:= n*Sum[Binomial[2*n,k]*(n-k)^(2*n-2), {k,0,n-1}] +Boole[n==1];
Table[A060279[n], {n,40}] (* G. C. Greubel, Nov 05 2024 *)

a(n) = n/2^n*sum(k=0, n, binomial(n, k)*(n-2*k)^(n-2)) \\ Michel Marcus, Jun 17 2013

def A060279(n): return n*sum( binomial(2*n,k)*(n-k)^(2*n-2) for k in range(n)) + int(n==1)
[A060279(n) for n in range(1,41)] # G. C. Greubel, Nov 05 2024

a(n) = (n/2^n)*Sum_{k=0..n} binomial(n, k)*(n-2*k)^(n-2).
a(n) = 2*n * A007106(n).
a(n) ~ sqrt(1+s^2) * s^(2*n-1) * 2^(2*n) * n^(2*n-1) / exp(2*n), where s = 1.5088795615383199289... is the root of the equation sqrt(1+s^2) = s*log(s+sqrt(1+s^2)). - Vaclav Kotesovec, Jan 23 2014

cp's OEIS Frontend

A058014 Number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.

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A143601 Number of labeled odd-degree trees with 2n+1 nodes.

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A370431 Expansion of e.g.f. S(x,k) satisfying S(x,k) = sinh( x*cosh(k*x*sqrt(1 + S(x,k)^2)) ), as a triangle read by rows.

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A370433 Expansion of e.g.f. T(x,k) satisfying T(x,k) = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T(x,k)^2)) ), as a triangle read by rows.

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A143600 E.g.f. satisfies: A(x) = exp(x*A(x)/A(-x)).

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A370331 Expansion of e.g.f. S(x,k) satisfying S(x,k) = sin( x*cos(k*x*sqrt(1 - S(x,k)^2)) ), as a triangle read by rows.

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A370333 Expansion of e.g.f. T(x,k) satisfying T(x,k) = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T(x,k)^2)) ), as a triangle read by rows.

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A370431 Expansion of e.g.f. S(x,k) satisfying S(x,k) = sinh( xcosh(kx*sqrt(1 + S(x,k)^2)) ), as a triangle read by rows.

A370433 Expansion of e.g.f. T(x,k) satisfying T(x,k) = (1/k) * sinh( kxcosh(xsqrt(1 + k^2T(x,k)^2)) ), as a triangle read by rows.

A370331 Expansion of e.g.f. S(x,k) satisfying S(x,k) = sin( xcos(kx*sqrt(1 - S(x,k)^2)) ), as a triangle read by rows.

A370333 Expansion of e.g.f. T(x,k) satisfying T(x,k) = (1/k) * sin( kxcos(xsqrt(1 - k^2T(x,k)^2)) ), as a triangle read by rows.

A372812 Expansion of e.g.f. S(x) satisfying S(x) = sinh( xcosh( 2x*sqrt(1 + S(x)^2) ) ), where a(n) is the coefficient of x^(2n+1)/(2n+1)! in S(x) for n >= 0.

A372814 Expansion of e.g.f. T(x) satisfying T(x) = (1/2) * sinh( 2xcosh( xsqrt(1 + 4T(x)^2) ) ), where a(n) is the coefficient of x^(2n+1)/(2n+1)! in T(x) for n >= 0.