cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A381282 Expansion of e.g.f. 1/(1 - x * cos(2*x)).

Original entry on oeis.org

1, 1, 2, -6, -72, -520, -1200, 24752, 516992, 5106816, 5287680, -998945024, -23719719936, -272471972864, 1326261594112, 149170761246720, 3843177252618240, 42752553478356992, -863092250325614592, -59317347865870139392, -1577115871098630307840, -13173264127625587851264
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A352252, A381283.
Cf. A185951.

Programs

  • PARI
    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!*(2*I)^(n-k)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-4)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * (2*i)^(n-k) * A185951(n,k), where i is the imaginary unit.

A385283 Expansion of e.g.f. 1/(1 - 2 * x * cos(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 3, -39, -775, -9045, -85813, -426447, 7321329, 325555155, 7786757011, 137053423881, 1388713844713, -21121997539461, -1827406866674085, -69034283067822495, -1852635543265039903, -30574875232261547613, 308376017794648053539, 54871741689019890859065
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A352252, A385284.
Cf. A001586, A235134, A380155, A385281.
Cf. A001147, A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*(2*I)^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * (2*i)^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.

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

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

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

Programs

  • Mathematica
    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}]
  • PARI
    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

Formula

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

A381344 Expansion of e.g.f. 1/( 1 - x * cosh(sqrt(2)*x) ).

Original entry on oeis.org

1, 1, 2, 12, 72, 500, 4560, 47936, 565376, 7572240, 112838400, 1844425792, 32910332928, 636463467328, 13251265570816, 295598326909440, 7034150340034560, 177843592245969152, 4760839037033054208, 134528586280018721792, 4001489050575059025920, 124973219149863342633984
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A352252, A381280, A381281, A381282, A381283, A381345.
Cf. A185951.

Programs

  • PARI
    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!*2^((n-k)/2)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 2^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * 2^((n-k)/2) * A185951(n,k).
a(n) ~ sqrt(Pi) * 2^(n/2 + 1) * n^(n + 1/2) / ((cosh(r) + r*sinh(r)) * exp(n) * r^(n+1)), where r = 0.95090803593755778120914299086438615849657408871... is the root of the equation r*cosh(r) = sqrt(2). - Vaclav Kotesovec, Apr 19 2025

A385284 Expansion of e.g.f. 1/(1 - 3 * x * cos(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 1, -152, -3515, -54080, -671363, -2823296, 199955305, 10101514240, 323321153881, 7583054076928, 80180394219757, -4570380001660928, -409907196093564395, -20705306119297925120, -748794938843475359663, -14289862480447260852224, 610587389113316064978481
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A352252, A385283.
Cf. A007559, A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*(3*I)^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * (3*i)^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.

A381208 Expansion of e.g.f. 1/(1 - x*cos(x))^2.

Original entry on oeis.org

1, 2, 6, 18, 48, 10, -1440, -17654, -153216, -1003950, -2787840, 58057538, 1483941888, 22381115354, 245730121728, 1455189928890, -18135147970560, -856283065534046, -19218870434267136, -306007541260257422, -2933654664287354880, 20552099782407258282, 1938717354581701951488
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A352252, A381209.
Cf. A185951.

Programs

  • PARI
    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+1)!*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} (k+1)! * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381345 Expansion of e.g.f. 1/( 1 - x * cos(sqrt(2)*x) ).

Original entry on oeis.org

1, 1, 2, 0, -24, -220, -1200, -2576, 52864, 1016208, 10909440, 57039488, -687971328, -26190716864, -450123634688, -4238375059200, 24514848522240, 2156422420074752, 54984136073084928, 799573460292407296, 42320889956270080, -425007017470737816576, -15563879892284330213376
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A352252, A381280, A381281, A381282, A381283, A381344.
Cf. A185951.

Programs

  • PARI
    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!*(-2)^((n-k)/2)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-2)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * (-2)^((n-k)/2) * A185951(n,k).
Previous Showing 11-17 of 17 results.

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