A327034 -id:A327034 - cp's OEIS frontend

A330046 Expansion of e.g.f. exp(x) / (1 - sinh(x)).

1, 2, 5, 17, 77, 437, 2975, 23627, 214457, 2189897, 24846395, 310095887, 4221990437, 62273111357, 989164604615, 16834483468547, 305604501324017, 5894522593612817, 120381876933435635, 2595103478745235607, 58887707028270711197, 1403084759749993342277

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

Binomial transform of A006154.

Cf. A000629, A006154, A009283, A327034, A330047.

nmax = 21; CoefficientList[Series[Exp[x]/(1 - Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = Sum_{k=0..n} binomial(n,k) * A006154(k).

a(n) ~ n! * (1 + 1/sqrt(2)) / (log(1 + sqrt(2)))^(n+1). - Vaclav Kotesovec, Dec 03 2019

A348587 Expansion of e.g.f. exp(x) / (2 - cos(x)).

Original entry on oeis.org

1, 1, 0, -2, 2, 26, -30, -622, 982, 25846, -50910, -1639142, 3874862, 147434366, -406614390, -17851478062, 56266545142, 2799621404086, -9927225631470, -552054087163382, 2175042302117822, 133686372253841006, -579383205000618150, -39002628245713951102

Offset: 0

Ilya Gutkovskiy, Oct 24 2021

sign

Cf. A003701, A094088, A327034, A348580.

nmax = 23; CoefficientList[Series[Exp[x]/(2 - Cos[x]), {x, 0, nmax}], x] Range[0, nmax]!

my(x='x+O('x^40)); Vec(serlaplace(exp(x)/(2-cos(x)))) \\ Michel Marcus, Oct 24 2021

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n,2*k) * A094088(k).

A214407 Triangle read by rows. The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1; k even} binomial(n, k) * p{k}(0)*(1 - n%2 + x^(n - k)).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 7, 0, 6, 0, 1, 0, 35, 0, 10, 0, 1, 121, 0, 105, 0, 15, 0, 1, 0, 847, 0, 245, 0, 21, 0, 1, 3907, 0, 3388, 0, 490, 0, 28, 0, 1, 0, 35163, 0, 10164, 0, 882, 0, 36, 0, 1, 202741, 0, 175815, 0, 25410, 0, 1470, 0, 45, 0, 1, 0

Offset: 0

Peter Luschny, Jul 16 2012

Matrix inverse of a signed variant of A119467.

			1
0, 1
1, 0, 1
0, 3, 0, 1
7, 0, 6, 0, 1
0, 35, 0, 10, 0, 1
121, 0, 105, 0, 15, 0, 1
0, 847, 0, 245, 0, 21, 0, 1
3907, 0, 3388, 0, 490, 0, 28, 0, 1

Cf. A119467, A327034 (row sums), A094088 (column 0).

@CachedFunction
def A214407_poly(n, x) :
    return 1 if n==0 else add(A214407_poly(k, 0)*binomial(n, k)*(x^(n-k)+1-n%2) for k in range(n)[::2])
def A214407_row(n) :
    R = PolynomialRing(ZZ, 'x')
    return R(A214407_poly(n,x)).coeffs()
for n in (0..8) : A214407_row(n)

T(n, k) = n! * [y^k] [x^n] (exp(x * y) / (2 - cosh(x))). - Peter Luschny, May 06 2023

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A330046 Expansion of e.g.f. exp(x) / (1 - sinh(x)).

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A348587 Expansion of e.g.f. exp(x) / (2 - cos(x)).

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A214407 Triangle read by rows. The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1; k even} binomial(n, k) * p{k}(0)*(1 - n%2 + x^(n - k)).

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