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 1-4 of 4 results.

A381504 Expansion of e.g.f. exp(-x/4) / (1-4*x)^(1/16).

Original entry on oeis.org

1, 0, 1, 8, 99, 1616, 32815, 797256, 22552873, 728069984, 26413495281, 1063820511080, 47098650935611, 2273501091042288, 118834339196361919, 6686552010270859496, 402975635704196998545, 25897425517232941658816, 1767875520978811381774753, 127753191169784612437640904
Offset: 0

Views

Author

Seiichi Manyama, Apr 23 2025

Keywords

Crossrefs

Cf. A000166, A383313, A381484.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x/4)/(1-4*x)^(1/16)))

Formula

a(n) = (-1)^n * n! * Sum_{k=0..n} (1/4)^(n-2*k) * binomial(-1/16,k)/(n-k)!.
a(n) = (n-1) * (4*a(n-1) + a(n-2)) for n > 1.
a(n) ~ sqrt(Pi) * 2^(2*n + 1/2) * n^(n - 7/16) / (Gamma(1/16) * exp(n + 1/16)). - Vaclav Kotesovec, Apr 23 2025

A381484 Expansion of e.g.f. exp(-x/3) / (1-3*x)^(1/9).

Original entry on oeis.org

1, 0, 1, 6, 57, 708, 10905, 200538, 4287633, 104507496, 2860291089, 86853807630, 2897638853769, 105357244427244, 4146601837761513, 175632278607964962, 7965651564924845985, 385161391574120046672, 19778647046883844762017, 1074979845580061777989014
Offset: 0

Views

Author

Seiichi Manyama, Apr 23 2025

Keywords

Crossrefs

Cf. A000166, A383313, A381504.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x/3)/(1-3*x)^(1/9)))

Formula

a(n) = (-1)^n * n! * Sum_{k=0..n} (1/3)^(n-2*k) * binomial(-1/9,k)/(n-k)!.
a(n) = (n-1) * (3*a(n-1) + a(n-2)) for n > 1.
a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 7/18) / (Gamma(1/9) * exp(n + 1/9)). - Vaclav Kotesovec, Apr 23 2025

A383314 Expansion of e.g.f. exp(-x/2) / (1-4*x)^(1/8).

Original entry on oeis.org

1, 0, 2, 16, 204, 3392, 69880, 1717824, 49077392, 1597961728, 58410015264, 2368359845120, 105492853521088, 5120497605295104, 269008689666893696, 15207860554294309888, 920541893947665404160, 59401332750388003782656, 4070589051420604880962048
Offset: 0

Views

Author

Seiichi Manyama, Apr 23 2025

Keywords

Crossrefs

Cf. A383313, A383315.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x/2)/(1-4*x)^(1/8)))

Formula

a(n) = (-1)^n * n! * Sum_{k=0..n} (1/2)^(n-3*k) * binomial(-1/8,k)/(n-k)!.
a(n) = 2*(n-1) * (2*a(n-1) + a(n-2)) for n > 1.
a(n) ~ sqrt(Pi) * 2^(2*n + 1/2) * n^(n - 3/8) / (Gamma(1/8) * exp(n + 1/8)). - Vaclav Kotesovec, Apr 23 2025

A383315 Expansion of e.g.f. exp(-x/2) / (1-6*x)^(1/12).

Original entry on oeis.org

1, 0, 3, 36, 675, 16632, 509085, 18626436, 793001097, 38511087120, 2101009734099, 127215916659540, 8465583820754907, 614101808094096744, 48230098800348987405, 4077120575169267005268, 369111206211249734907345, 35630377583888099367357984, 3653123185073359871950788963
Offset: 0

Views

Author

Seiichi Manyama, Apr 23 2025

Keywords

Crossrefs

Cf. A383313, A383314.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x/2)/(1-6*x)^(1/12)))

Formula

a(n) = (-1)^n * n! * Sum_{k=0..n} 3^k * (1/2)^(n-2*k) * binomial(-1/12,k)/(n-k)!.
a(n) = 3*(n-1) * (2*a(n-1) + a(n-2)) for n > 1.
a(n) ~ Pi * (2 - sqrt(3))^(1/4) * 2^(n + 1/2) * 3^(n - 3/8) * n^(n - 5/12) / (Gamma(1/3) * Gamma(1/4) * exp(n + 1/12)). - Vaclav Kotesovec, Apr 23 2025
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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