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A126156 E.g.f. sqrt(sec(sqrt(2)*x)), showing coefficients of only the even powers of x. 8
1, 1, 7, 139, 5473, 357721, 34988647, 4784061619, 871335013633, 203906055033841, 59618325600871687, 21297483077038703899, 9127322584507530151393, 4621897483978366951337161, 2730069675607609356178641127 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Previous name was: Column 0 and row sums of symmetric triangle A126155.

This is the sqrt 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

LINKS

Table of n, a(n) for n=0..14.

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

It appears that sum(n>=0, a(n)*x^(2*n)/(2*n)! ) = sqrt( sec(sqrt(2)*x)). - David Callan, Jan 03 2011

G.f.: 1/(1-x/(1-6x/(1-15x/(1-28x/(1-45x/(1-66x/(1-91x/(1-... or

1/U(0), U(k)=1-x(k+1)(2k+1)/U(k+1); (continued fraction). - 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

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

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 *)

CROSSREFS

Cf. A126155; diagonals: A126157, A126158.

Sequence in context: A137463 A221375 A190195 * A082162 A238692 A085708

Adjacent sequences:  A126153 A126154 A126155 * A126157 A126158 A126159

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

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Last modified September 2 13:04 EDT 2014. Contains 246357 sequences.