A003717 -id:A003717 - cp's OEIS frontend

A006229 Expansion of e.g.f. exp( tan x ).

1, 1, 1, 3, 9, 37, 177, 959, 6097, 41641, 325249, 2693691, 24807321, 241586893, 2558036145, 28607094455, 342232522657, 4315903789009, 57569080467073, 807258131578995, 11879658510739497, 183184249105857781, 2948163649552594737, 49548882107764546223

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259, Sum_{k} T(n,k).
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..480 (terms 0..100 from T. D. Noe)
J. Shallit, Letter to N. J. A. Sloane, May 1975
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Row sums of A059419 and unsigned A111593.

Cf. A003711, A003717, A000182, A047691, A047692.

function A006229_list(len::Int)
    len <= 0 && return BigInt[]
    T = zeros(BigInt, len, len); T[1,1] = 1
    S = Array(BigInt, len); S[1] = 1
    for n in 2:len
        T[n,n] = 1
        for k in 2:n-1 T[n,k] = T[n-1,k-1] + k*(k-1)*T[n-1,k+1] end
        S[n] = sum(T[n,k] for k in 2:n)
    end
S end
println(A006229_list(24)) # Peter Luschny, Apr 27 2017

With[{nn=30},CoefficientList[Series[Exp[Tan[x]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 04 2011 *)

a(n):=sum(if oddp(n+k) then 0 else (-1)^((n+k)/2)*sum(j!/k!*stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1, k-1), j, k, n), k, 1, n); /* Vladimir Kruchinin, Aug 05 2010 */

E.g.f.: exp(tan(x)).
a(n) = sum(if oddp(n+k) then 0 else (-1)^((n+k)/2)*sum(j!/k!*stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1,k-1),j,k,n),k,1,n), n>0. - Vladimir Kruchinin, Aug 05 2010
E.g.f.: 1 + tan(x)/T(0), where T(k) = 4*k+1 - tan(x)/(2 + tan(x)/(4*k+3 - tan(x)/(2 + tan(x)/T(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
a(n) = Sum_{i=0..(n-1)/2} binomial(n-1,2*i)*z(i+1)*a(n-2*i-1), a(0)=1, where z(n) is tangent (or "zag") numbers (A000182). - Vladimir Kruchinin, Mar 04 2015 [corrected by Jason Yuen, Dec 29 2024]

More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001

A296854 Expansion of e.g.f. sinh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, 18, 227, 4565, 126648, 4620805, 213569269, 12165013026, 835868220455, 68093897815361, 6483538063860336, 712877916658802713, 89586864207214060057, 12753583150716684461970, 2040805972702652020364603, 364567588100855831300341565

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			sinh(x*tan(x/2)) = x^2/2! + x^4/4! + 18*x^6/6! + 227*x^8/8! + 4565*x^10/10! + ...

Cf. A001469, A003717, A009611, A110501, A296839, A296841, A296842, A296853, A296856.

nmax = 17; Table[(CoefficientList[Series[Sinh[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] sinh(x*tan(x/2)).

A296680 Expansion of e.g.f. arcsin(arctanh(x)) (odd powers only).

Original entry on oeis.org

1, 3, 53, 2359, 198953, 27412011, 5625656541, 1613676694239, 617477049181521, 304167421243513683, 187546541676182230149, 141512355477854459198343, 128265950128144233675269241, 137512081213377707268891639675, 172108297920263623816775456321325

Offset: 0

Ilya Gutkovskiy, Dec 18 2017

nonn

			arcsin(arctanh(x)) = x/1! + 3*x^3/3! + 53*x^5/5! + 2359*x^7/7! + 198953*x^9/9! + 27412011*x^11/11! + ...

Robert Israel, Table of n, a(n) for n = 0..215

Cf. A001818, A003717, A010050, A012134, A296464, A296466, A296679.

S:= series(arcsin(arctanh(x)),x,52):
seq(coeff(S,x,n)*n!,n=1..51,2); # Robert Israel, Dec 18 2017

nmax = 15; Table[(CoefficientList[Series[ArcSin[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 15; Table[(CoefficientList[Series[-I Log[(I/2) (Log[1 + x] - Log[1 - x]) + Sqrt[1 - (Log[1 + x] - Log[1 - x])^2/4]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

E.g.f.: arcsinh(arctan(x)) (odd powers only, absolute values).

E.g.f.: -i*log((i/2)*(log(1 + x) - log(1 - x)) + sqrt(1 - (log(1 + x) - log(1 - x))^2/4)), where i is the imaginary unit (odd powers only).

A013522 Numerator of [x^(2n+1)] in the Taylor expansion sinh(cosec(x)-cotan(x))= x/2 +x^3/16 +37x^5/3840 +137x^7/92160 +41641x^9/185794560 + 3887x^11/117964800 +...

Original entry on oeis.org

1, 1, 37, 137, 41641, 3887, 241586893, 5721418891, 4315903789009, 2832484672207, 183184249105857781, 2154299222076719401, 1431144441595717024523, 386845480523042818420133, 21349170171172632123182767, 38112676874301043070814698873, 25659732417088795005806537367241

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

Apparently a bisection of A047691.

The numerators in the e.g.f. of x/2, sinh(cosec(x)-cotan(x)) = x/(2^1*1!) +3*x^3/(2^3*3!) +37*x^5/(2^5*5!) +959*x^7/(2^7*7!) +41641*x^9/(2^9*9!)+.. are apparently covered by the absolute values of A003717.

G. C. Greubel, Table of n, a(n) for n = 0..200

Numerator[Take[CoefficientList[Series[Sinh[Csc[x] - Cot[x]], {x,0,45}], x], {2, -1, 2}]] (* G. C. Greubel, Nov 12 2016 *)

Name edited by R. J. Mathar, Dec 19 2011

A098146 First odd semiprime > 10^n.

Original entry on oeis.org

9, 15, 111, 1003, 10001, 100001, 1000001, 10000001, 100000001, 1000000013, 10000000003, 100000000007, 1000000000007, 10000000000015, 100000000000013, 1000000000000003, 10000000000000003, 100000000000000015

Offset: 0

Hugo Pfoertner, Aug 28 2004

nonn

			a(0)=9 because 9=3*3 is the first odd semiprime following 10^0=1.
a(13) = 10000000000015 = 5*2000000000003.

Dario Alpern, Factorization using the Elliptic Curve Method.

Cf. A046315 (odd semiprimes), A098147(n)=a(n)-10^n continuation of this sequence, A003717 (smallest n-digit prime).

osp[n_]:=Module[{k=1},While[PrimeOmega[n+k]!=2,k=k+2];n+k]; Join[{9}, Table[osp[10^i],{i,20}]] (* Harvey P. Dale, Jan 17 2012 *)

print1(9,","); for(n=1,10,forstep(i=10^n+1,10^(n+1)-1,2,f=factor(i); ms=matsize(f); if((ms[1]==1&&f[1,2]==2)||(ms[1]==2&&f[1,2]==1&&f[2,2]==1),print1(i,","); break))) /* Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 21 2006 */

from sympy import factorint, nextprime
def is_semiprime(n): return sum(e for e in factorint(n).values()) == 2
def next_odd_semiprime(n):
    nxt = n + 1 + n%2
    while not is_semiprime(nxt): nxt += 2
    return nxt
def a(n): return next_odd_semiprime(10**n)
print([a(n) for n in range(20)]) # Michael S. Branicky, Sep 15 2021

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A006229 Expansion of e.g.f. exp( tan x ).

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A296854 Expansion of e.g.f. sinh(x*tan(x/2)) (even powers only).

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A296680 Expansion of e.g.f. arcsin(arctanh(x)) (odd powers only).

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A013522 Numerator of [x^(2n+1)] in the Taylor expansion sinh(cosec(x)-cotan(x))= x/2 +x^3/16 +37*x^5/3840 +137*x^7/92160 +41641*x^9/185794560 + 3887*x^11/117964800 +...

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A098146 First odd semiprime > 10^n.

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A013522 Numerator of [x^(2n+1)] in the Taylor expansion sinh(cosec(x)-cotan(x))= x/2 +x^3/16 +37x^5/3840 +137x^7/92160 +41641x^9/185794560 + 3887x^11/117964800 +...