A126156 Expansion of e.g.f. sqrt(sec(sqrt(2)*x)), showing coefficients of only the even powers of x.
1, 1, 7, 139, 5473, 357721, 34988647, 4784061619, 871335013633, 203906055033841, 59618325600871687, 21297483077038703899, 9127322584507530151393, 4621897483978366951337161, 2730069675607609356178641127, 1860452328661957054823447670979, 1448802510679254790311316267306753
Offset: 0
Keywords
Examples
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! + ...
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.
Programs
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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
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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 *) -
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 */
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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 */
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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
def A126156(n): return A126155(n, 0) print([A126156(n) for n in range(17)]) # Peter Luschny, Dec 14 2023
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
Extensions
New name based on a comment of David Callan, Peter Luschny, May 07 2014
Comments