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.

A326984 Coefficients in asymptotic expansion of sequence A309619.

Original entry on oeis.org

1, 0, 1, 1, 3, 13, 57, 271, 1467, 8905, 58965, 420331, 3212391, 26227477, 227640033, 2090172631, 20222758995, 205524856129, 2188159483341, 24344716477411, 282390978550239, 3408195810080461, 42719427069801369, 555174137978970511, 7469189351830156683
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 10 2019

Keywords

Examples

			A309619(n) / n! ~ 1 + 1/n^2 + 1/n^3 + 3/n^4 + 13/n^5 + 57/n^6 + ...

Links

Crossrefs

Cf. A309619, A293266.

A309618 a(n) = Sum_{k=0..floor(n/2)} k! * 2^k * (n - 2*k)!.

Original entry on oeis.org

1, 1, 4, 8, 36, 140, 832, 5376, 42432, 374592, 3720960, 40694784, 486679296, 6310114560, 88168366080, 1320468480000, 21101183631360, 358354687426560, 6444941507297280, 122367252835860480, 2445878526994022400, 51337143210820239360, 1128918790687649955840
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 10 2019

Keywords

Links

Crossrefs

Cf. A000142, A003149, A107713, A110143, A309619.

Programs

  • Mathematica
    Table[Sum[k!*2^k*(n-2*k)!, {k, 0, Floor[n/2]}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Sum[k!*x^k, {k, 0, nmax}] * Sum[k!*2^k*x^(2 k), {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: B(x)*B(2*x^2), where B(x) is g.f. of A000142.
a(n) ~ n! * (1 + 2/n^2 + 2/n^3 + 10/n^4 + 50/n^5 + 250/n^6 + 1442/n^7 + 9514/n^8 + 68882/n^9 + 539098/n^10 + ...), for coefficients see A326983.

A339984 G.f.: g(x) * g(x^2), where g(x) is the g.f. of A000081.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 5, 13, 26, 65, 147, 369, 899, 2298, 5851, 15261, 39945, 105948, 282504, 759480, 2052027, 5576017, 15216998, 41705762, 114715503, 316611401, 876466003, 2433091773, 6771462322, 18889829555, 52809592990, 147935027381, 415182991401, 1167251435240
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 25 2020

Keywords

Links

Crossrefs

Cf. A000081, A217781, A309619, A339985.

Programs

  • Mathematica
    max = 30; A000081 = Rest[Cases[ Import["https://oeis.org/A000081/b000081.txt", "Table"], {, }][[All, 2]]]; g81 = Sum[A000081[[k]]*x^k, {k, 1, max}]; g81x2 = Sum[A000081[[k]]*x^(2*k), {k, 1, max}]; CoefficientList[Series[g81 * g81x2, {x, 0, max}], x]

Formula

a(n) ~ A339986 * A051491^n / n^(3/2).

A339515 a(n) = Sum_{k=0..floor(n/3)} k! * (n - 3*k)!.

Original entry on oeis.org

1, 1, 2, 7, 25, 122, 728, 5066, 40444, 363618, 3633894, 39957372, 479365980, 6230659848, 87218289408, 1308154099944, 20929024197336, 355774686465840, 6403682340295200, 121666035674658960, 2433257870201802720, 51097347163646718480, 1124122414761046131120
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 07 2020

Keywords

Crossrefs

Cf. A096161, A309619.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Sum[k!*x^k, {k, 0, nmax}] * Sum[k!*x^(3*k), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Sum[k!*(n - 3*k)!, {k, 0, Floor[n/3]}], {n, 0, 25}]
  • PARI
    a(n) = sum(k=0, n\3, k! * (n - 3*k)!); \\ Michel Marcus, Dec 08 2020

Formula

G.f.: B(x)*B(x^3), where B(x) is g.f. of A000142.
a(n) ~ n! * (1 + 1/n^3 + 3/n^4 + 7/n^5 + 17/n^6 + 61/n^7 + 343/n^8 + 2233/n^9 + 14373/n^10 + ...).
Showing 1-4 of 4 results.

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