A279010 -id:A279010 - 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

A279009 Alternating Jacobsthal triangle A_{-2}(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, -2, 0, 1, -2, -2, -1, 1, 4, 0, -1, -2, 1, 4, 4, 1, 1, -3, 1, -8, 0, 3, 0, 4, -4, 1, -8, -8, -3, 3, -4, 8, -5, 1, 16, 0, -5, -6, 7, -12, 13, -6, 1, 16, 16, 5, 1, -13, 19, -25, 19, -7, 1, -32, 0, 11, 4, 14, -32, 44, -44, 26, -8, 1

Offset: 0

N. J. A. Sloane, Dec 07 2016

			Triangle begins:
1,
1,   1,
-2,  0,  1,
-2, -2, -1,  1,
4,   0, -1, -2,   1,
4,   4,  1,  1,  -3,   1,
-8,  0,  3,  0,   4, - 4,   1,
-8, -8, -3,  3,  -4,   8,  -5,   1,
16,  0, -5, -6,   7, -12,  13,  -6,  1,
16, 16,  5,  1, -13,  19, -25,  19, -7,  1,
-32, 0, 11,  4,  14, -32,  44, -44, 26, -8, 1,
...

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

Cf. A112468, A279006, A279010.

a[n_, 0] := (-2)^Floor[n/2]; a[n_, n_] = 1; a[n_, k_] /; 0 <= k <= n := a[n, k] = a[n-1, k-1] - a[n-1, k]; a[, ] = 0;
Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 05 2018 *)

cp's OEIS Frontend

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

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