A009744 -id:A009744 - cp's OEIS frontend

A024235 Expansion of e.g.f. tan(x)*sin(x)/2 (even powers only).

0, 1, 2, 31, 692, 25261, 1351382, 99680491, 9695756072, 1202439837721, 185185594118762, 34674437196568951, 7757267081778543452, 2043536254646561946181, 626129820701814932734142, 220771946624511552276841411, 88759695789769644718332394832

Offset: 0

R. H. Hardin

From Peter Bala, Nov 10 2016: (Start)
This sequence gives the coefficients in an asymptotic expansion related to the constant Pi/8. Recall the Madhava-Gregory-Leibniz series Pi/4 = Sum_{k = 1..inf} (-1)^(k-1)/(2*k - 1). Borwein et al. gave an asymptotic expansion for the tails of this series: Pi/2 - 2*Sum_{k = 1..N/2} (-1)^(k-1)/(2*k - 1) ~ 1/N - 1/N^3 + 5/N^5 - 61/N^7 + ..., where N is an integer divisible by 4 and the sequence of unsigned coefficients [1, 1, 5, 61,...] is the sequence of Euler numbers A000364.
Similarly, we have the series representation Pi/8 = Sum_{k = 1..inf} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)): using the approach of Borwein et al. we can show the associated asymptotic expansion for the tails of the series is Pi/4 - 2*Sum_{k = 1..N/2} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)) ~ -1/N^3 + 2/N^5 - 31/N^7 + 692/N^9 - ..., where N is divisible by 4 and where the sequence of unsigned coefficients [1, 2, 31, 692,...] forms the present sequence. A numerical example is given below. Cf. A278080 and A278195. (End)

			tan(x)*sin(x)/2 = 1/2*x^2 + 1/12*x^4 + 31/720*x^6 + 173/10080*x^8 + ...
From _Peter Bala_, Nov 10 2016: (Start)
Asymptotic expansion at N = 100000.
The truncated series 2*Sum_{k = 1..N/2} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)) = 0.78539816339744(9)309615660(6)4581987(603) 104929(1657)84377... to 50 digits. The bracketed digits show where this decimal expansion differs from that of Pi/4. The numbers -1, 2, -31, 692 must be added to the bracketed numbers to give the correct decimal expansion to 50 digits: Pi/4 = 0.78539816339744(8)309615660(8)4581987(572)104929(2349)84377.... (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..50
J. M. Borwein, P. B. Borwein, K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
Eric Weisstein's World of Mathematics, Euler Polynomial.

Cf. A009744, A000364, A004174, A019675, A278080, A278195.

A000364 := proc(n)
   abs(euler(2*n));
end proc:
seq(1/2*(A000364(n) - (-1)^n), n = 0..20); # Peter Bala, Nov 10 2016

With[{nn=30},Take[CoefficientList[Series[Tan[x]*Sin[x]/2,{x,0,nn}], x]Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Apr 27 2012 *)

G.f.: 1/2*(G(0) - 1/(1+x)) where G(k) = 1 - x*(2*k+1)^2/(1 - x*(2*k+2)^2/G(k+1) );  (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 09 2013
a(n) ~ (2*n)! * (2/Pi)^(2*n+1). - Vaclav Kotesovec, Jan 23 2015
From Peter Bala, Nov 10 2016: (Start)
a(n) = 1/2*(A000364(n) - (-1)^n).
a(n) = 1/8*(-4)^n*( -E(2*n,3/2) + 2*E(2*n,1/2) - E(2*n,-1/2) ), where E(n,x) is the Euler polynomial of order n.
G.f. 1/2!*sin^2(x)/cos(x) = x^2/2! + 2*x^4/4! + 31*x^6/6! + 692*x^8/8! + ....
O.g.f. for a signed version of the sequence: Sum_{n >= 0} ( 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n, k)/((1 - (2*k - 1)*x)*(1 - (2*k + 1)*x)*(1 - (2*k + 3)*x)) ) = 1 - 2*x^2 + 31*x^4 - 692*x^6 + .... (End)

Extended and signs tested Mar 15 1997.

More terms from Harvey P. Dale, Apr 27 2012

A099023 Diagonal of Euler-Seidel matrix with start sequence e.g.f. 1-tanh(x).

Original entry on oeis.org

1, -1, 4, -46, 1024, -36976, 1965664, -144361456, 13997185024, -1731678144256, 266182076161024, -49763143319190016, 11118629668610842624, -2925890822304510631936, 895658946905031792553984

Offset: 0

Ralf Stephan, Sep 23 2004

sign

T(2n,n), where T is A008280 (signed).

Peter Luschny, An old operation on sequences: the Seidel transform

Cf. A000657, A008280, A029582, A009744.

A099023List[n_] := Module[{e, dim, m, k}, dim = 2 n; e[0, 0] = 1; For[m = 1, m <= dim - 1, m++, If[EvenQ[m], e[m, 0] = 1; For[k = m - 1, k >= -1, k--, e[k, m - k] = e[k + 1, m - k - 1] - e[k, m - k - 1]], e[0, m] = 1; For[k = 1, k <= m + 1, k++, e[k, m - k] = e[k - 1, m - k + 1] + e[k - 1, m - k]]]]; Table[e[k, k], {k, 0, (dim + 1)/2 - 1}]];
A099023List[15] (* Jean-François Alcover, Jun 11 2019, after Peter Luschny *)

# Variant of an algorithm of L. Seidel (1877).
def A099023_list(n) :
    dim = 2*n; E = matrix(ZZ, dim); E[0,0] = 1
    for m in (1..dim-1) :
        if m % 2 == 0 :
            E[m,0] = 1;
            for k in range(m-1,-1,-1) :
                E[k,m-k] = E[k+1,m-k-1] - E[k,m-k-1]
        else :
            E[0,m] = 1;
            for k in range(1,m+1,1) :
                E[k,m-k] = E[k-1,m-k+1] + E[k-1,m-k]
    return [E[k,k] for k in range((dim+1)//2)]
# Peter Luschny, Jul 14 2012

|a(n)| = A000657(n) - Sean A. Irvine, Dec 22 2010
G.f.: 1/G(0) where G(k) = 1 + x*(k+1)*(4*k+1)/(1 + x*(k+1)*(4*k+3)/G(k+1) ) ;  (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 05 2013
G.f.: G(0)/(1+x), where G(k) = 1 - x^2*(k+1)^2*(4*k+1)*(4*k+3)/( x^2*(k+1)^2*(4*k+1)*(4*k+3) - (1 + x*(8*k^2+4*k+1))*(1 + x*(8*k^2+20*k+13))/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 01 2014

A029582 E.g.f. sin(x) + cos(x) + tan(x).

Original entry on oeis.org

1, 2, -1, 1, 1, 17, -1, 271, 1, 7937, -1, 353791, 1, 22368257, -1, 1903757311, 1, 209865342977, -1, 29088885112831, 1, 4951498053124097, -1, 1015423886506852351, 1, 246921480190207983617, -1, 70251601603943959887871, 1, 23119184187809597841473537

Offset: 0

N. J. A. Sloane

sign

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A009744, A099023.

nn = 30; Range[0, nn]! CoefficientList[Series[Tan[x] + Sin[x] + Cos[x], {x, 0, nn}], x] (* T. D. Noe, Jul 16 2012 *)

# Variant of an algorithm of L. Seidel (1877).
def A029582_list(n) :
    dim = 2*n; E = matrix(ZZ, dim); E[0, 0] = 1
    for m in (1..dim-1) :
        if m % 2 == 0 :
            E[m, 0] = 1;
            for k in range(m-1, -1, -1) :
                E[k, m-k] = E[k+1, m-k-1] - E[k, m-k-1]
        else :
            E[0, m] = 1;
            for k in range(1, m+1, 1) :
                E[k, m-k] = E[k-1, m-k+1] + E[k-1, m-k]
    return [(-1)^(k//2)*E[k,0] for k in range(dim)]
A029582_list(15)  # Peter Luschny, Jul 14 2012

G.f.: (1+x)/(1+x^2)+x/T(0) where T(k)= 1 - (k+1)*(k+2)*x^2/T(k+1) ; (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 12 2013

G.f.: (1+x)/(1+x^2)+x/G(0) where G(k) = 1 - 2*x^2*(4*k^2+4*k+1) - 4*x^4*(k+1)^2*(4*k^2+8*k+3)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 12 2013

A296462 Expansion of e.g.f. arcsin(x)*arctanh(x) (even powers only).

Original entry on oeis.org

0, 2, 12, 238, 9912, 708282, 77392260, 12002011110, 2507167177200, 678724656721650, 231129344455890300, 96694934804540934750, 48752132066414189721000, 29154453671147281799726250, 20403607225475633039372992500, 16520371586328834323725749873750, 15322889489994265975004588078700000

Offset: 0

Ilya Gutkovskiy, Dec 13 2017

nonn

			arcsin(x)*arctanh(x) = 2*x^2/2! + 12*x^4/4! + 238*x^6/6! + 9912*x^8/8! + 708282*x^10/10! + ...

Cf. A001818, A009744, A009747, A010050, A012723, A296463.

nmax = 16; Table[(CoefficientList[Series[ArcSin[x] ArcTanh[x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 16; Table[(CoefficientList[Series[I (Log[1 - x] - Log[1 + x]) Log[I x + Sqrt[1 - x^2]]/2, {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

E.g.f.: arcsinh(x)*arctan(x) (even powers only, absolute values).
E.g.f.: i*(log(1 - x) - log(1 + x))*log(i*x + sqrt(1 - x^2))/2, where i is the imaginary unit (even powers only).
a(n) ~ Pi * (2*n-1)! / 2. - Vaclav Kotesovec, Dec 13 2017

A296463 Expansion of e.g.f. arcsinh(x)*arctanh(x) (even powers only).

Original entry on oeis.org

0, 2, 4, 158, 3624, 427482, 29665260, 6948032310, 991515848400, 383952670412850, 93532380775766100, 53913667654307868750, 20087427376748637675000, 16096655588343149442026250, 8531309209053208518037597500, 9057367559484733295974741323750, 6486329752640392315697926589700000

Offset: 0

Ilya Gutkovskiy, Dec 13 2017

nonn

			arcsinh(x)*arctanh(x) = 2*x^2/2! + 4*x^4/4! + 158*x^6/6! + 3624*x^8/8! + 427482*x^10/10! + ..

Cf. A001818, A009744, A009747, A010050, A012752, A296462.

nmax = 16; Table[(CoefficientList[Series[ArcSinh[x] ArcTanh[x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 16; Table[(CoefficientList[Series[(Log[1 + x] - Log[1 - x]) Log[x + Sqrt[1 + x^2]]/2, {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

E.g.f.: arcsin(x)*arctan(x) (even powers only, absolute values).
E.g.f.: (log(1 + x) - log(1 - x))*log(x + sqrt(1 + x^2))/2 (even powers only).
a(n) ~ (2*n-1)! * log(1+sqrt(2)) * (1 - (-1)^n * sqrt(Pi) / (4 * log(1+sqrt(2)) * sqrt(n))). - Vaclav Kotesovec, Dec 13 2017

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A024235 Expansion of e.g.f. tan(x)*sin(x)/2 (even powers only).

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A099023 Diagonal of Euler-Seidel matrix with start sequence e.g.f. 1-tanh(x).

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A029582 E.g.f. sin(x) + cos(x) + tan(x).

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A296462 Expansion of e.g.f. arcsin(x)*arctanh(x) (even powers only).

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A296463 Expansion of e.g.f. arcsinh(x)*arctanh(x) (even powers only).

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