OFFSET
0,3
COMMENTS
Previous name was: Column 0 and row sums of symmetric triangle A126155.
This is the square root of the Euler numbers (A122045) with respect to the Cauchy type product as described by J. Singh (see link and the second Maple program) normalized by 2^n. A241885 shows the corresponding sqrt of the Bernoulli numbers. - Peter Luschny, May 07 2014
REFERENCES
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 366.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..200
Peter Bala, A triangle for calculating A126156
Alain Connes, Caterina Consani and Henri Moscovici, Zeta zeros and prolate wave operators, arXiv:2310.18423 [math.NT], Oct 2023, p.31.
Denis S. Grebenkov, Vittoria Sposini, Ralf Metzler, Gleb Oshanin, and Flavio Seno, Exact distributions of the maximum and range of random diffusivity processes, New J. Phys. (2021) Vol. 23, 023014.
Jitender Singh, On an arithmetic convolution, arXiv:1402.0065 [math.NT], 2014.
FORMULA
a(n) = Sum_{k=0..n} A087736(n,k)*3^(n-k). - Philippe Deléham, Jul 17 2007
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = sqrt(sec(sqrt(2)*x)). - David Callan, Jan 03 2011
E.g.f. satisfies: A(x) = exp( Integral Integral A(x)^4 dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, May 30 2015
E.g.f. satisfies: A(x) = exp( Integral A(x)^2 * Integral 1/A(x)^2 dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, Jun 02 2015
G.f.: 1/(1-x/(1-6*x/(1-15*x/(1-28*x/(1-45*x/(1-66*x/(1-91*x/(1-... or 1/U(0) where U(k) = 1-x*(k+1)*(2*k+1)/U(k+1); (continued fraction). [See Wall.] - Sergei N. Gladkovskii, Oct 31 2011
G.f.: 1/U(0) where U(k) = 1 - (4*k+1)*(4*k+2)*x/(2 - (4*k+3)*(4*k+4)*x/ U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 24 2012
G.f.: 1/G(0) where G(k) = 1 -x*(k+1)*(2*k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 11 2013
G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)*(k+1)/( x*(2*k+1)*(k+1) - 1/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 09 2013
a(n) ~ 2^(5*n+2) * n^(2*n) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, Jul 13 2014
a(n) = (1/(4*n))*Sum_{k=1..n} binomial(2*n,2*k)*((2^(2*k)-1)*2^(3*k)*(-1)^((k-1))*Bernoulli(2*k)*a(n-k)), a(0)=1. - Vladimir Kruchinin, Feb 25 2015
a(n) = Sum_{k=1..n} a(n-k)*binomial(2*n,2*k)*(k/(2*n)-1)*(-2)^k, a(0)=1. - Tani Akinari, Sep 11 2023
For n > 0, a(n) = -Sum_{j=0..n} Sum_{k=0..floor(j/2)} (2*n+1)!*(2*k-j)^(2*n)/(n!*(2*j+1)*(n-j)!*k!*(j-k)!*(-2)^(n+j-1)). - Tani Akinari, Sep 28 2023
EXAMPLE
E.g.f.: A(x) = 1 + x^2/2! + 7*x^4/4! + 139*x^6/6! + 5473*x^8/8! + 357721*x^10/10! + ...
where the logarithm begins:
log(A(x)) = x^2/2! + 4*x^4/4! + 64*x^6/6! + 2176*x^8/8! + 126976*x^10/10! + 11321344*x^12/12! + ...
compare the logarithm to
A(x)^4 = 1 + 4*x^2/2! + 64*x^4/4! + 2176*x^6/6! + 126976*x^8/8! + 11321344*x^10/10! + ...
MAPLE
A126156 := proc(n)
sqrt(sec(sqrt(2)*z)) ;
coeftayl(%, z=0, 2*n) ;
%*(2*n)! ;
end;
seq(A126156(n), n=0..10) ; # Sergei N. Gladkovskii, Oct 31 2011
g := proc(f, n) option remember; local g0, m; g0 := sqrt(f(0));
if n=0 then g0 else if n=1 then 0 else add(binomial(n, m)*g(f, m)* g(f, n-m), m=1..n-1) fi; (f(n)-%)/(2*g0) fi end:
a := n -> (-2)^n*g(euler, 2*n);
seq(a(n), n=0..14); # Peter Luschny, May 07 2014
# Alternative: an algorithm as described by Peter Bala, see also A365672:
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then
T(n, k-1) else (n - k + 1) * (2 * (n - k) + 1) * T(n, k - 1) + T(n - 1, k)
fi fi end:
a := n -> T(n, n): seq(a(n), n = 0..14); # Peter Luschny, Sep 29 2023
MATHEMATICA
a[n_] := SeriesCoefficient[ Sqrt[ Sec[ Sqrt[2]*x]], {x, 0, 2 n}]*(2*n)!; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Nov 29 2013, after Sergei N. Gladkovskii *)
PROG
(Maxima)
a(n):=if n=0 then 1 else 1/(4*n)*sum(binomial(2*n, 2*k)*((2^(2*k)-1)*2^(3*k)*(-1)^((k-1))*bern(2*k)*a(n-k)), k, 1, n); /* Vladimir Kruchinin, Feb 25 2015 */
(Maxima)
a[n]:=if n=0 then 1 else sum(a[n-k]*binomial(2*n, 2*k)*(k/(2*n)-1)*(-2)^k, k, 1, n);
makelist(a[n], n, 0, 30); /* Tani Akinari, Sep 11 2023 */
(PARI) /* E.g.f. A(x) = exp( Integral^2 A(x)^4 dx^2 ): */
{a(n)=local(A=1+x*O(x)); for(i=1, n, A=exp(intformal(intformal(A^4 + x*O(x^(2*n))))) ); (2*n)!*polcoeff(A, 2*n, x)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = local(A=1+x); for(i=1, n, A = exp( intformal( A^2 * intformal( 1/A^2 + x*O(x^n)) ) ) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(2*n), ", "))
(PARI) {a(n)=-(n<1)-sum(j=0, n, sum(k=0, j/2, (2*n+1)!*(2*k-j)^(2*n)/(n!*(2*j+1)*(n-j)!*k!*(j-k)!*(-2)^(n+j-1))))}; /* Tani Akinari, Sep 28 2023 */
(SageMath)
print([A126156(n) for n in range(17)]) # Peter Luschny, Dec 14 2023
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 20 2006
EXTENSIONS
New name based on a comment of David Callan, Peter Luschny, May 07 2014
STATUS
approved