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.

A360339 a(n) = coefficient of x^n*y^(2*n+1)/n! in log( Sum_{n>=0} (n + y)^(3*n) * x^n/n! ).

Original entry on oeis.org

1, 6, 99, 2832, 117405, 6423408, 438143391, 35869775616, 3430351996569, 375544727136000, 46333978359977979, 6362713275564589056, 962689133095843525749, 159139760744994666835968, 28539360163037720058960375, 5518961894002049077780611072, 1144859158421455331276272257201
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2023

Keywords

Examples

			E.g.f.: A(x) = x + 6*x^2/2! + 99*x^3/3! + 2832*x^4/4! + 117405*x^5/5! + 6423408*x^6/6! + 438143391*x^7/7! + 35869775616*x^8/8! + ... + a(n)*x^n/n! + ...
where a(n) equals the coefficient of y^(2*n+1)*x^n/n! in the series given by
log( Sum_{n>=0} (n + y)^(3*n) * x^n/n! ) = (y^3 + 3*y^2 + 3*y + 1)*x + (6*y^5 + 45*y^4 + 140*y^3 + 225*y^2 + 186*y + 63)*x^2/2! + (99*y^7 + 1305*y^6 + 7722*y^5 + 26514*y^4 + 56844*y^3 + 75780*y^2 + 57915*y + 19493)*x^3/3! + (2832*y^9 + 56214*y^8 + 521784*y^7 + 2965716*y^6 + 11339280*y^5 + 30131946*y^4 + 55424512*y^3 + 67771380*y^2 + 49792368*y + 16686958)*x^4/4! + (117405*y^11 + 3214647*y^10 + 42201705*y^9 + 349928235*y^8 + 2030468625*y^7 + 8627152275*y^6 + 27284511927*y^5 + 63980788365*y^4 + 108602299435*y^3 + 126629082945*y^2 + 90978438315*y + 30421607649)*x^5/5! + ...
Exponentiation yields the e.g.f. of A266482:
exp(A(x)) = 1 + x + 7*x^2/2! + 118*x^3/3! + 3373*x^4/4! + 139096*x^5/5! + 7565779*x^6/6! + 513277024*x^7/7! + 41820455065*x^8/8! + 3982842285184*x^9/9! + 434457816912991*x^10/10! + ... + A266483(n)*x^n/n! + ...
which equals
lim_{N->oo} [ Sum_{n>=0} (N + n)^(3*n) * (x/N^2)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 3, 33, 708, 23481, 1070568, 62591913, 4483721952, ...].

Links

Crossrefs

Cf. A266482, A360340, A360341, A359926.

Programs

  • PARI
    /* Using logarithmic formula */
{a(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(3*m) *x^m/m! ) +x*O(x^n) ), n, x), 2*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)^(3*n) * (x/N^2)^n/n! ).
(2) a(n) = [x^n*y^(2*n+1)/n!] log( Sum_{n>=0} (n + y)^(3*n) * x^n/n! ).
a(n) ~ c * d^n * n! / n^(5/2), where d = (3/2) * (3 + sqrt(6)) * exp(3 - sqrt(6)) = 14.175247991325192557234088913125084764719990898660219459... and c = sqrt((3 - sqrt(6))/Pi)/4 = 0.1046520596183180437324097699670683850916674939335504... - Vaclav Kotesovec, Feb 12 2023, updated Mar 20 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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