A352251 -id:A352251 - cp's OEIS frontend

A352252 Expansion of e.g.f. 1 / (1 - x * cos(x)).

1, 1, 2, 3, 0, -55, -480, -3157, -15232, -16623, 898560, 16316179, 194574336, 1666248025, 5418649600, -170157839685, -5164467978240, -92955464490463, -1188910801354752, -7329026447550685, 157257042777866240, 7516793832172469481, 187200588993188069376

Offset: 0

Ilya Gutkovskiy, Mar 09 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..456

Cf. A000111, A007289, A009189, A094088, A205571, A302397, A352250, A352251.

nmax = 22; CoefficientList[Series[1/(1 - x Cos[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^k Binomial[n, 2 k + 1] (2 k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 22}]

my(x='x+O('x^30)); Vec(serlaplace(1 / (1 - x * cos(x)))) \\ Michel Marcus, Mar 10 2022

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*I^(n-k)*a185951(n, k)); \\ Seiichi Manyama, Feb 17 2025

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

a(n) = Sum_{k=0..n} k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 17 2025

A352250 Expansion of e.g.f. 1 / (1 - x * sin(x)) (even powers only).

Original entry on oeis.org

1, 2, 20, 486, 21944, 1591210, 169207092, 24808395262, 4796420822384, 1182349445882706, 361939981107422060, 134705596642758848806, 59900689507397744253096, 31365504832631796986962426, 19102102945852191813235300004, 13387748268024668296590660222030

Offset: 0

Ilya Gutkovskiy, Mar 09 2022

nonn

Cf. A000111, A000364, A009214, A205571, A210657, A302397, A352251, A352252.

nmax = 30; Take[CoefficientList[Series[1/(1 - x Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = 2 Sum[(-1)^(k + 1) Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sin(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

a(0) = 1; a(n) = 2 * Sum_{k=1..n} (-1)^(k+1) * binomial(2*n,2*k) * k * a(n-k).

A352253 Expansion of e.g.f. 1 / (1 - x * sinh(x) / 2) (even powers only).

Original entry on oeis.org

1, 1, 8, 153, 5492, 316625, 26774622, 3121729709, 479962730648, 94087054172673, 22904161764512570, 6778870099212235805, 2397161662661680925364, 998186321121004312238513, 483430830256916593106991782, 269435322393253822641626419725, 171224984800186115316322226731952

Offset: 0

Ilya Gutkovskiy, Mar 09 2022

nonn

Cf. A006153, A006154, A094088, A205571, A352251, A352254.

nmax = 32; Take[CoefficientList[Series[1/(1 - x Sinh[x]/2), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sinh(x)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n,2*k) * k * a(n-k).

cp's OEIS Frontend

A352252 Expansion of e.g.f. 1 / (1 - x * cos(x)).

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A352250 Expansion of e.g.f. 1 / (1 - x * sin(x)) (even powers only).

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A352253 Expansion of e.g.f. 1 / (1 - x * sinh(x) / 2) (even powers only).

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Author

Keywords

Crossrefs

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Mathematica

PARI

Formula