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A126156 Expansion of e.g.f. sqrt(sec(sqrt(2)*x)), showing coefficients of only the even powers of x.

%I A126156 #101 Dec 14 2023 10:10:38
%S A126156 1,1,7,139,5473,357721,34988647,4784061619,871335013633,
%T A126156 203906055033841,59618325600871687,21297483077038703899,
%U A126156 9127322584507530151393,4621897483978366951337161,2730069675607609356178641127,1860452328661957054823447670979,1448802510679254790311316267306753
%N A126156 Expansion of e.g.f. sqrt(sec(sqrt(2)*x)), showing coefficients of only the even powers of x.
%C A126156 Previous name was: Column 0 and row sums of symmetric triangle A126155.
%C A126156 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
%D A126156 H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 366.
%H A126156 G. C. Greubel, <a href="/A126156/b126156.txt">Table of n, a(n) for n = 0..200</a>
%H A126156 Peter Bala, <a href="/A126156/a126156.pdf">A triangle for calculating A126156</a>
%H A126156 Alain Connes, Caterina Consani and Henri Moscovici, <a href="https://doi.org/10.48550/arXiv.2310.18423">Zeta zeros and prolate wave operators</a>, arXiv:2310.18423 [math.NT], Oct 2023, p.31.
%H A126156 Denis S. Grebenkov, Vittoria Sposini, Ralf Metzler, Gleb Oshanin, and Flavio Seno, <a href="https://doi.org/10.1088/1367-2630/abd313">Exact distributions of the maximum and range of random diffusivity processes</a>, New J. Phys. (2021) Vol. 23, 023014.
%H A126156 Jitender Singh, <a href="http://arxiv.org/abs/1402.0065">On an arithmetic convolution</a>, arXiv:1402.0065 [math.NT], 2014.
%F A126156 a(n) = Sum_{k=0..n} A087736(n,k)*3^(n-k). - _Philippe Deléham_, Jul 17 2007
%F A126156 E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = sqrt(sec(sqrt(2)*x)). - _David Callan_, Jan 03 2011
%F A126156 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
%F A126156 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
%F A126156 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
%F A126156 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
%F A126156 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
%F A126156 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
%F A126156 a(n) ~ 2^(5*n+2) * n^(2*n) / (exp(2*n) * Pi^(2*n+1/2)). - _Vaclav Kotesovec_, Jul 13 2014
%F A126156 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
%F A126156 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
%F A126156 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
%e A126156 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! + ...
%e A126156 where the logarithm begins:
%e A126156 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! + ...
%e A126156 compare the logarithm to
%e A126156 A(x)^4 = 1 + 4*x^2/2! + 64*x^4/4! + 2176*x^6/6! + 126976*x^8/8! + 11321344*x^10/10! + ...
%p A126156 A126156 := proc(n)
%p A126156         sqrt(sec(sqrt(2)*z)) ;
%p A126156         coeftayl(%,z=0,2*n) ;
%p A126156         %*(2*n)! ;
%p A126156 end;
%p A126156 seq(A126156(n),n=0..10) ; # _Sergei N. Gladkovskii_, Oct 31 2011
%p A126156 g := proc(f, n) option remember; local g0, m; g0 := sqrt(f(0));
%p A126156 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:
%p A126156 a := n -> (-2)^n*g(euler, 2*n);
%p A126156 seq(a(n), n=0..14); # _Peter Luschny_, May 07 2014
%p A126156 # Alternative: an algorithm as described by _Peter Bala_, see also A365672:
%p A126156 T := proc(n, k) option remember; if k = 0 then 1 else if k = n then
%p A126156 T(n, k-1) else (n - k + 1) * (2 * (n - k) + 1) * T(n, k - 1) + T(n - 1, k)
%p A126156 fi fi end:
%p A126156 a := n -> T(n, n): seq(a(n), n = 0..14);  # _Peter Luschny_, Sep 29 2023
%t A126156 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_ *)
%o A126156 (Maxima)
%o A126156 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 */
%o A126156 (Maxima)
%o A126156 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);
%o A126156 makelist(a[n],n,0,30); /* _Tani Akinari_, Sep 11 2023 */
%o A126156 (PARI) /* E.g.f. A(x) = exp( Integral^2 A(x)^4 dx^2 ): */
%o A126156 {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)}
%o A126156 for(n=0,20,print1(a(n),", "))
%o A126156 (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)}
%o A126156 for(n=0,20,print1(a(2*n),", "))
%o A126156 (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 */
%o A126156 (SageMath)
%o A126156 def A126156(n): return A126155(n, 0)
%o A126156 print([A126156(n) for n in range(17)])  # _Peter Luschny_, Dec 14 2023
%Y A126156 Diagonals: A126157, A126158.
%Y A126156 Cf. A126155, A365672.
%K A126156 nonn
%O A126156 0,3
%A A126156 _Paul D. Hanna_, Dec 20 2006
%E A126156 New name based on a comment of _David Callan_, _Peter Luschny_, May 07 2014