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 1-2 of 2 results.

A360340 a(n) = coefficient of x^n*y^(3*n+1)/n! in log( Sum_{n>=0} (n + y)^(4*n) * x^n/n! ).

Original entry on oeis.org

1, 8, 180, 7072, 403960, 30504384, 2874754624, 325376606720, 43039201623552, 6519192650444800, 1113116854379470336, 211577875772377853952, 44316053154112985589760, 10142584803973143241244672, 2518533121682934512363520000, 674412844392686430750000676864
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2023

Keywords

Examples

			E.g.f.: A(x) = x + 8*x^2/2! + 180*x^3/3! + 7072*x^4/4! + 403960*x^5/5! + 30504384*x^6/6! + 2874754624*x^7/7! + 325376606720*x^8/8! + ... + a(n)*x^n/n! + ...
where a(n) equals the coefficient of y^(3*n+1)*x^n/n! in the series given by
log( Sum_{n>=0} (n + y)^(4*n) * x^n/n! ) = (y^4 + 4*y^3 + 6*y^2 + 4*y + 1)*x + (8*y^7 + 84*y^6 + 392*y^5 + 1050*y^4 + 1736*y^3 + 1764*y^2 + 1016*y + 255)*x^2/2! + (180*y^10 + 3392*y^9 + 30138*y^8 + 165768*y^7 + 622692*y^6 + 1662072*y^5 + 3178509*y^4 + 4282316*y^3 + 3875094*y^2 + 2119644*y + 530675)*x^3/3! + (7072*y^13 + 203056*y^12 + 2832672*y^11 + 25357888*y^10 + 161977312*y^9 + 776565264*y^8 + 2862877120*y^7 + 8183026480*y^6 + 18063131520*y^5 + 30301902248*y^4 + 37428709376*y^3 + 32144205840*y^2 + 17161326976*y + 4292647990)*x^4/4! + ...
Exponentiation yields the e.g.f. of A266483:
exp(A(x)) = 1 + x + 9*x^2/2! + 205*x^3/3! + 8033*x^4/4! + 456561*x^5/5! + 34307545*x^6/6! + 3219222301*x^7/7! + 363018204225*x^8/8! + ... + A266483(n)*x^n/n! + ...
which equals
lim_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 4, 60, 1768, 80792, 5084064, 410679232, 40672075840, 4782133513728, ...].

Links

Crossrefs

Cf. A266526, A360339, A360341, A359926.

Programs

  • PARI
    /* Using logarithmic formula */
{a(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(4*m) *x^m/m! ) +x*O(x^n) ), n, x), 3*n+1, y)}
for(n=1, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! may be defined as follows.
(1) A(x) = Limit_{N->oo} (1/N) * log( Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ).
(2) a(n) = [x^n*y^(3*n+1)/n!] log( Sum_{n>=0} (n + y)^(4*n) * x^n/n! ).
a(n) ~ c * d^n * n! / n^(5/2), where d = (16/9) * (3 + 2*sqrt(3)) * exp(2*(2 - sqrt(3))) = 19.6391804025535695723623649299349856005700748518058428... and c = (sqrt(3) - 1) / (6*sqrt(Pi)) = 0.06883590670968059270411496568598690388462... - Vaclav Kotesovec, Feb 12 2023, updated Mar 17 2024

A360341 a(n) = coefficient of x^n*y^(3*n+1)/n! in log( Sum_{n>=0} (n + y)^(5*n) * x^n/n! ).

Original entry on oeis.org

1, 10, 285, 14240, 1036225, 99774720, 11995938325, 1732780710400, 292580972777025, 56581144474976000, 12335796889894262125, 2994228576573719040000, 800930404887937807458625, 234113078032084301026816000, 74248479783538967821383793125, 25394786139647229685682094080000
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2023

Keywords

Examples

			E.g.f.: A(x) = x + 10*x^2/2! + 285*x^3/3! + 14240*x^4/4! + 1036225*x^5/5! + 99774720*x^6/6! + 11995938325*x^7/7! + 1732780710400*x^8/8! + ... + a(n)*x^n/n! + ...
where a(n) equals the coefficient of y^(4*n+1)*x^n/n! in the series given by
log( Sum_{n>=0} (n + y)^(5*n) * x^n/n! ) = (y^5 + 5*y^4 + 10*y^3 + 10*y^2 + 5*y + 1)*x + (10*y^9 + 135*y^8 + 840*y^7 + 3150*y^6 + 7812*y^5 + 13230*y^4 + 15240*y^3 + 11475*y^2 + 5110*y + 1023)*x^2/2! + (285*y^13 + 6985*y^12 + 82800*y^11 + 626640*y^10 + 3365015*y^9 + 13480875*y^8 + 41269545*y^7 + 97340225*y^6 + 176218089*y^5 + 241023105*y^4 + 241403365*y^3 + 167262045*y^2 + 71713845*y + 14345837)*x^3/3! + (14240*y^17 + 535150*y^16 + 9965360*y^15 + 121806600*y^14 + 1090732800*y^13 + 7563031080*y^12 + 41870604200*y^11 + 188252006020*y^10 + 693127766960*y^9 + 2094270509580*y^8 + 5176075514880*y^7 + 10375810342800*y^6 + 16622405553984*y^5 + 20792525880990*y^4 + 19576849364160*y^3 + 13053873999580*y^2 + 5496952909520*y + 1099451098702)*x^4/4! + ...
Exponentiation yields the e.g.f. of A266484:
exp(A(x)) = 1 + x + 11*x^2/2! + 316*x^3/3! + 15741*x^4/4! + 1140376*x^5/5! + 109350271*x^6/6! + 13100626176*x^7/7! + 1886686497401*x^8/8! + ... + A266484(n)*x^n/n! + ...
which equals
lim_{N->oo} [ Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 5, 95, 3560, 207245, 16629120, 1713705475, 216597588800, ...].

Links

Crossrefs

Cf. A266484, A360339, A360340.

Programs

  • PARI
    /* Using logarithmic formula */
{a(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(5*m) *x^m/m! ) +x*O(x^n) ), n, x), 4*n+1, y)}
for(n=1, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! may be defined as follows.
(1) A(x) = Limit_{N->oo} (1/N) * log( Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ).
(2) a(n) = [x^n*y^(3*n+1)/n!] log( Sum_{n>=0} (n + y)^(5*n) * x^n/n! ).
a(n) ~ c * d^n * n! / n^(5/2), where d = (25/16) * (5 + 2*sqrt(5)) * exp(5 - 2*sqrt(5)) = 25.090908742294025045771061662375185533388200826641029119554... and c = 1/(8*sqrt((1 + 2/sqrt(5))*Pi)) = 0.05123846578813482717849518499100286... - Vaclav Kotesovec, Feb 12 2023, updated Mar 20 2024
Showing 1-2 of 2 results.

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