A172388 -id:A172388 - cp's OEIS frontend

A340451 E.g.f.: Sum_{n>=0} x^n * cosh(2^n*x) / n!.

1, 1, 2, 13, 98, 721, 7682, 165313, 4816898, 154732801, 7052328962, 587435092993, 67748952539138, 9011561121239041, 1692739935456460802, 557257804202631217153, 255875811615404841762818, 138681207656726645785559041, 105975684493162347867458764802

Offset: 0

Paul D. Hanna, Jan 12 2021

nonn

a(n) = A047863(n) - A340452(n) for n >= 0, in which A047863 gives the number of labeled graphs with 2-colored nodes when connected only to nodes of a different color.

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 13*x^3/3! + 98*x^4/4! + 721*x^5/5! + 7682*x^6/6! + 165313*x^7/7! + 4816898*x^8/8! + 154732801*x^9/9! + ...
where
A(x) = cosh(x) + x*cosh(2*x) + x^2*cosh(2^2*x)/2! + x^3*cosh(2^3*x)/3! + x^4*cosh(2^4*x)/4! + x^5*cosh(2^5*x)/5! + ...
also
A(x) = exp(x) + x^2*exp(2^2*x)/2! + x^4*exp(2^4*x)/4! + x^6*exp(2^6*x)/6! + x^8*exp(2^8*x)/8! + ...

Cf. A047863, A172388, A340452.

{a(n) = my(A = sum(m=0, n, x^m/m! * cosh(2^m*x +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A = sum(m=0, n\2, x^(2*m)/(2*m)! * exp(2^(2*m)*x +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: Sum_{n>=0} x^n * cosh(2^n*x) / n!.

E.g.f.: Sum_{n>=0} x^(2*n) * exp(4^n*x) / (2*n)!.

A340452 E.g.f.: Sum_{n>=0} x^n * sinh(2^n*x) / n!.

Original entry on oeis.org

1, 4, 13, 64, 721, 10624, 165313, 3672064, 154732801, 8959043584, 587435092993, 54484881424384, 9011561121239041, 2072928719458238464, 557257804202631217153, 211345500389721452314624, 138681207656726645785559041, 127174792727050845731397566464

Offset: 1

Paul D. Hanna, Jan 12 2021

nonn

a(n) = A047863(n) - A340451(n) for n >= 0, in which A047863 gives the number of labeled graphs with 2-colored nodes when connected only to nodes of a different color.

			E.g.f.: A(x) = x + 4*x^2/2! + 13*x^3/3! + 64*x^4/4! + 721*x^5/5! + 10624*x^6/6! + 165313*x^7/7! + 3672064*x^8/8! + 154732801*x^9/9! + ...
where
A(x) = sinh(x) + x*sinh(2*x) + x^2*sinh(2^2*x)/2! + x^3*sinh(2^3*x)/3! + x^4*sinh(2^4*x)/4! + x^5*sinh(2^5*x)/5! + ...
also
A(x) = x*exp(2*x) + x^3*exp(2^3*x)/3! + x^5*exp(2^5*x)/5! + x^7*exp(2^7*x)/7! + x^9*exp(2^9*x)/9! + ...

Cf. A047863, A172388, A340451.

{a(n) = my(A = sum(m=0, n, x^m/m! * sinh(2^m*x +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))

{a(n) = my(A = sum(m=0, n\2+1, x^(2*m+1)/(2*m+1)! * exp(2^(2*m+1)*x +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))

E.g.f.: Sum_{n>=0} x^n * sinh(2^n*x) / n!.

E.g.f.: Sum_{n>=0} x^(2*n+1) * exp(2^(2*n+1)*x) / (2*n+1)!.

cp's OEIS Frontend

A340451 E.g.f.: Sum_{n>=0} x^n * cosh(2^n*x) / n!.

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A340452 E.g.f.: Sum_{n>=0} x^n * sinh(2^n*x) / n!.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula