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.

Showing 1-3 of 3 results.

A381283 Expansion of e.g.f. 1/(1 - x * cos(3*x)).

Original entry on oeis.org

1, 1, 2, -21, -192, -1095, 7200, 243747, 3088512, 1360881, -874437120, -21701765349, -186175604736, 5870711879721, 292185085151232, 5507319584787795, -38951106749890560, -6402114772676575263, -212680600451474522112, -1602903494245708491957, 197042528380347210792960
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, A381282.
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!*(3*I)^(n-k)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-9)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * (3*i)^(n-k) * A185951(n,k), where i is the imaginary unit.

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

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).
Showing 1-3 of 3 results.

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