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.

A266485 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + 2*n)^(2*n) * (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 9, 193, 6929, 356001, 24004825, 2012327521, 202156421409, 23701550853313, 3179302948594601, 480443117415138945, 80788534008942735409, 14965275494082095616097, 3028424508967743713615481, 664790043100841638943719201, 157352199248412053285546165825, 39950540529265571984889165180801
Offset: 0

Views

Author

Paul D. Hanna, Dec 30 2015

Keywords

Comments

Compare to: Limit_{N->oo} [ Sum_{n>=0} (N + n)^n * x^n/n! ]^(1/N) = Sum_{n>=0} (n+1)^(n-1) * x^n/n!.
Related limits (Paul D. Hanna, Jan 20 2023):
exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
W(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n)^n * (x/N)^n/n! ]^(1/N), where W(x) = LambertW(-x)/(-x).

Examples

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 193*x^3/3! + 6929*x^4/4! + 356001*x^5/5! + 24004825*x^6/6! + 2012327521*x^7/7! + 202156421409*x^8/8! + 23701550853313*x^9/9! + 3179302948594601*x^10/10! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N+2)^2*(x/N) + (N+4)^4*(x/N)^2/2! + (N+6)^6*(x/N)^3/3! + (N+8)^8*(x/N)^4/4! + (N+10)^10*(x/N)^5/5! + (N+12)^12*(x/N)^6/6! +...]^(1/N).
The logarithm of the g.f. A(x) begins (_Paul D. Hanna_, Jan 20 2023):
(a) log(A(x)) = x + 8*x^2/2! + 168*x^3/3! + 6016*x^4/4! + 309760*x^5/5! + 20957184*x^6/6! + 1762991104*x^7/7! + 177690607616*x^8/8! + ... + A359926(n)*x^n/n! + ...
where A359926(n) = [x^n*y^(n+1)/n!] (1/4) * log( Sum_{n>=0} (n + 2*y)^(2*n) * x^n/n! );
that is, the coefficient of x^n/n! in the logarithm of e.g.f A(x) equals the coefficient of y^(n+1)*x^n/n! in the series given by
(b) (1/4) * log( Sum_{n>=0} (n + 2*y)^(2*n) * x^n/n! ) = (y^2 + y + 1/4)*x + (8*y^3 + 18*y^2 + 14*y + 15/4)*x^2/2! + (168*y^4 + 632*y^3 + 933*y^2 + 639*y + 683/4)*x^3/3! + (6016*y^5 + 33088*y^4 + 76480*y^3 + 92680*y^2 + 58720*y + 31019/2)*x^4/4! + ...

Links

Crossrefs

Cf. A266481, A266482, A266483, A266484, A266486, A266487, A359926, A359927, A319147, A318633, A319834.

Programs

  • PARI
    /* Informal listing of terms 0..30 */
\p300
P(n) = sum(k=0,32, (n+2*k)^(2*k) * x^k/k! +O(x^31))
Vec( round( serlaplace( subst(P(10^100)^(1/10^100),x,x/10^100) )*1.) )
  • PARI
    /* Using formula for the logarithm of g.f. A(x) Paul D. Hanna, Jan 20 2023 */
{L(n) = (1/4) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + 2*y)^(2*m) *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following (Paul D. Hanna, Jan 20 2023):
(1) A(x) = lim_{N->oo} [ Sum_{n>=0} (N + 2*n)^(2*n) * (x/N)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} A359926(n)*x^n/n! ), where A359926(n) = (1/4) * [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n + 2*y)^(2*n) *x^n/n! ).
a(n) ~ c * d^n * n^(n-2), where d = 4*(1 + sqrt(2)) * exp(1 - sqrt(2)) = 6.3818267815342167443903123351857161682971406064645602440616... and c = sqrt(1 - 1/sqrt(2))/2 * exp(3/2 - sqrt(2)) = 0.294836494691148677397464568534316405253091784834436235... - Vaclav Kotesovec, Jan 21 2023, updated Mar 17 2024

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

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

A359918 a(n) = coefficient of x^n*y^(n+1)/n! in (1/2) * log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! ).

Original entry on oeis.org

1, 2, 21, 304, 6985, 205056, 7607509, 337188608, 17495079921, 1038495001600, 69496455755221, 5176052539987968, 424783071501394489, 38087843235679268864, 3704990294840345047125, 388631778963216211050496, 43729459820175064700435041, 5254332451028464517449777152
Offset: 1

Views

Author

Paul D. Hanna, Jan 21 2023

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 21*x^3/3! + 304*x^4/4! + 6985*x^5/5! + 205056*x^6/6! + 7607509*x^7/7! + 337188608*x^8/8! + 17495079921*x^9/9! + 1038495001600*x^10/10! + ...
Exponentiation yields the e.g.f. of A359917:
exp(A(x)) = 1 + x + 3*x^2/2! + 28*x^3/3! + 413*x^4/4! + 9216*x^5/5! + 268327*x^6/6! + 9831424*x^7/7! + 432251577*x^8/8! +...+ A359917(n)*x^n/n! + ...
which equals
lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 1, 7, 76, 1397, 34176, 1086787, 42148576, 1943897769, 103849500160, ...].

Links

Crossrefs

Cf. A359917, A359926, A359928, A319834, A318634.

Programs

  • PARI
    {a(n) = (1/2) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m^2 + m*y + 2*y^2)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}
for(n=1, 30, print1(a(n), ", "))

Formula

a(n) ~ c * d^n * n! / n^(5/2), where d = 7.68892218919697462312... and c = 0.1314019396717313039... - Vaclav Kotesovec, Mar 21 2024
Showing 1-4 of 4 results.

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