A279009 -id:A279009 - cp's OEIS frontend

A089171 Numerators of series coefficients of 1/(1 + cosh(sqrt(x))).

1, -1, 1, -17, 31, -691, 5461, -929569, 3202291, -221930581, 4722116521, -56963745931, 14717667114151, -2093660879252671, 86125672563201181, -129848163681107301953, 868320396104950823611, -209390615747646519456961, 14129659550745551130667441

Offset: 0

Wouter Meeussen, Dec 07 2003

Unsigned version is equal to A002425 up to n=11, but differs beyond that point.

Unsigned version: numerators of series coefficients of 1/(1 + cos(sqrt(x))); see Mathematica. - Clark Kimberling, Dec 06 2016

N. J. A. Sloane, Table of n, a(n) for n = 0..299

Cf. A002425, A050970, A002430, A046990.
Cf. A000182, A198631, A279009, A279010.
Cf. A276592, A279370.

with(numtheory): c := n->(2^(2*n)-1)*bernoulli(2*n)/(2*n)!; seq(numer(c(n)),n=1..20); # C. Ronaldo

Numerator[CoefficientList[Series[1/(1+Cosh[Sqrt[x]]), {x, 0, 24}], x]]
Numerator[CoefficientList[Series[1/(1+Cos[Sqrt[x]]), {x, 0, 30}], x]]
(* unsigned version, Clark Kimberling, Dec 06 2016 *)

a(n) = numerator(c(n+1)) where c(n)=(2^(2*n)-1)*B(2*n)/(2*n)!, B(k) denotes the k-th Bernoulli number. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004
Numerators of expansion of cosec(x)-cot(x) = 1/2*x+1/4*x^3/3!+1/2*x^5/5!+17/8*x^7/7!+31/2*x^9/9!+... - Ralf Stephan, Dec 21 2004 (Comment was applied to wrong entry, corrected by Alessandro Musesti (musesti(AT)gmail.com), Nov 02 2007)
E.g.f.: 1/sin(x)-cot(x). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: x/G(0); G(k) = 4*k+2-x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: (1+x/(x-2*Q(0)))/2; Q(k) = 8*k+2+x/(1+(2*k+1)*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: x/(x+Q(0)); Q(k) = x+(x^2)/((4*k+1)*(4*k+2)-(4*k+1)*(4*k+2)/(1+(4*k+3)*(4*k+4)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: T(0)/2, where T(k) = 1 - x^2/(x^2 - (4*k+2)*(4*k+6)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 12 2013
Aerated, these are the numerators of the Taylor series coefficients of 2 * tanh(x/2) (cf. A000182 and A198631). - Tom Copeland, Oct 19 2016

A279010 Alternating Jacobsthal triangle A_3(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 3, -1, 1, 9, 0, 4, -2, 1, 9, 9, -4, 6, -3, 1, 27, 0, 13, -10, 9, -4, 1, 27, 27, -13, 23, -19, 13, -5, 1, 81, 0, 40, -36, 42, -32, 18, -6, 1, 81, 81, -40, 76, -78, 74, -50, 24, -7, 1, 243, 0, 121, -116, 154, -152, 124, -74, 31, -8, 1

Offset: 0

N. J. A. Sloane, Dec 07 2016

			Triangle begins:
    1;
    1,  1;
    3,  0,   1;
    3,  3,  -1,    1;
    9,  0,   4,   -2,   1;
    9,  9,  -4,    6,  -3,    1;
   27,  0,  13,  -10,   9,   -4,   1;
   27, 27, -13,   23, -19,   13,  -5,   1;
   81,  0,  40,  -36,  42,  -32,  18,  -6,  1;
   81, 81, -40,   76, -78,   74, -50,  24, -7,  1;
  243,  0, 121, -116, 154, -152, 124, -74, 31, -8, 1;
  ...

Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

If initial column is omitted, this is very like the Riordan matrix A191582.

Cf. A112468, A279006, A279009.

A[n_, 0] := 3^Floor[n/2];
A[n_, k_] /; (k<0 || t>n) = 0;
A[n_, n_] = 1;
A[n_, k_] := A[n, k] = A[n-1, k-1] - A[n-1, k];
Table[A[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)

A279110 Denominators of coefficients in expansion of 2/(1 + cos(sqrt(x))).

Original entry on oeis.org

1, 4, 24, 2880, 40320, 7257600, 479001600, 697426329600, 20922789888000, 12804747411456000, 2432902008176640000, 264470759477084160000, 620448401733239439360000, 806582922253211271168000000, 304888344611713860501504000000, 4244045756995056938180935680000000

Offset: 0

Clark Kimberling, Dec 07 2016

			1/1 + (1/4)x + (1/24)x^2 + (17/2880)x^3 + ... ; 1/1, 1/4, 1/24, 17/2880, 31/40320, ... = |A089171|/A279110.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A089171, A279109.

z = 26; t = CoefficientList[Series[2/(1 + Cos[Sqrt[x]]), {x, 0, z}], x];
Numerator[t]   (* A089171, unsigned *)
Denominator[t] (* A279110 *)

2*A279009.

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A089171 Numerators of series coefficients of 1/(1 + cosh(sqrt(x))).

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A279010 Alternating Jacobsthal triangle A_3(n,k) read by rows.

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A279110 Denominators of coefficients in expansion of 2/(1 + cos(sqrt(x))).

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Mathematica

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