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 1-7 of 7 results.

A266481 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 5, 55, 993, 25501, 857773, 35850795, 1795564865, 104972371417, 7022842421301, 529428563641759, 44421725002096225, 4106744812439019765, 414834196219620026333, 45462732300569936279251, 5373006006732947705188737, 681229881246574750274962225, 92237589983019368975021777125, 13283769418970268811752725081607, 2027649185923009220298941142143201, 326999803592314489529958494308640461, 55558592280735155060861740192416874125
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!.
Conjecture: a(p*n) = 1 (mod p) for n>=0 and all prime p.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 993*x^4/4! + 25501*x^5/5! + 857773*x^6/6! + 35850795*x^7/7! + 1795564865*x^8/8! + 104972371417*x^9/9! + 7022842421301*x^10/10! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N+1)^2*(x/N) + (N+2)^4*(x/N)^2/2! + (N+3)^6*(x/N)^3/3! + (N+4)^8*(x/N)^4/4! + (N+5)^10*(x/N)^5/5! + (N+6)^12*(x/N)^6/6! +...]^(1/N).
RELATED SERIES.
The following limit exists:
G(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ] / A(x)^N
where
G(x) = 1 + 2*x + 22*x^2/2! + 432*x^3/3! + 12220*x^4/4! + 451480*x^5/5! + 20591784*x^6/6! + 1117635008*x^7/7! + 70348179472*x^8/8! + 5037843612960*x^9/9! + 404453425948000*x^10/10! +...+ A266522(n)*x^n/n! +...
Logarithm of the g.f. A(x) begins:
Log(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 752*x^4/4! + 19360*x^5/5! + 654912*x^6/6! + 27546736*x^7/7! + 1388207872*x^8/8! + 81621893376*x^9/9! + 5488951731200*x^10/10! +...+ A266526(n)*x^n/n! +...
and forms a diagonal in the triangles A266521 and A266488.

Links

Crossrefs

Cf. A266482, A266483, A266484, A266485, A266486, A266487, A266488, A266522, A266526.

Programs

  • PARI
    {A266526(n) = n! * polcoeff( polcoeff( log( sum(m=0,n+1, (m + 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, A266526(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* Informal listing of terms 0..30 */
\p100
P(n) = sum(k=0,31, (n+k)^(2*k) * x^k/k! +O(x^31))
Vec(round( serlaplace( subst(P(10^100)^(1/10^100),x,x/10^100) )*1.) )

Formula

E.g.f. exp( Sum_{n>=0} A266526(n)*x^n/n! ), where A266526(n) = [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n + y)^(2*n) * x^n/n! ).
a(n) ~ c * d^n * n^(n-2), where d = 2*(1 + sqrt(2)) * exp(1 - sqrt(2)) = 3.19091339076710837219515616759285808414857..., c = sqrt(1 - 1/sqrt(2)) * exp(3 - 2*sqrt(2)) = 0.642492128663019850313957348436... . - Vaclav Kotesovec, Jan 01 2016, updated Mar 17 2024

A266483 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 9, 205, 8033, 456561, 34307545, 3219222301, 363018204225, 47866764942721, 7230829461286121, 1231746006983485005, 233652055492688836129, 48852757000944980067505, 11163401061821489604439737, 2768164393136241898192002781, 740339555234437428570144337025, 212438189627800855103688740374401, 65104233055709355841104275116309705, 21223353839635626633833547837080498509, 7333306933167926737746819644785091452641
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!.

Examples

			E.g.f.: 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! + 47866764942721*x^9/9! + 7230829461286121*x^10/10! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N+1)^4*(x/N^3) + (N+2)^8*(x/N^3)^2/2! + (N+3)^12*(x/N^3)^3/3! + (N+4)^16*(x/N^3)^4/4! + (N+5)^20*(x/N^3)^5/5! + (N+6)^24*(x/N^3)^6/6! +...]^(1/N).

Links

Crossrefs

Cf. A266481, A266482, A266484, A266485, A266486.

Programs

  • PARI
    /* Informal listing of terms 0..30 */
\p400
P(n) = sum(k=0,32, (n+k)^(4*k) * x^k/k! +O(x^32))
Vec(round(serlaplace( subst(P(10^100)^(1/10^100),x,x/10^300) )*1.) )
  • PARI
    /* Using logarithmic formual */
{L(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)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 29 2023

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.
(1) A(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} L(n)*x^n/n! ), where L(n) = [x^n*y^(3*n+1)/n!] log( Sum_{n>=0} (n + y)^(4*n) * x^n/n! ). - Paul D. Hanna, Jan 29 2023
a(n) ~ 2^(3*n + 1/2) * (1 + sqrt(3))^(2*n-1) * exp((3-2*sqrt(3))*n - 4*sqrt(3) + 7) * n^(n-2) / 3^(3*n/2 + 1). - Vaclav Kotesovec, Mar 20 2024

A266482 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(3*n) * (x/N^2)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 7, 118, 3373, 139096, 7565779, 513277024, 41820455065, 3982842285184, 434457816912991, 53434112376345856, 7317518431787267653, 1104465712210096168960, 182183636400541105459627, 32609250878782525222260736, 6295153043394143761311198769, 1303848990485145459272159297536, 288415207140946760926622987982775, 67863051757810284274576363569872896, 16924929956887283486906002826128780381, 4459845456377312896416211474995205636096
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!.

Examples

			E.g.f.: 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! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N+1)^3*(x/N^2) + (N+2)^6*(x/N^2)^2/2! + (N+3)^9*(x/N^2)^3/3! + (N+4)^12*(x/N^2)^4/4! + (N+5)^15*(x/N^2)^5/5! + (N+6)^18*(x/N^2)^6/6! +...]^(1/N).

Links

Crossrefs

Cf. A266481, A266483, A266484, A266485, A266486, A266523, A266524, A266525.

Programs

  • PARI
    /* Informal listing of terms 0..30 */
\p200
P(n) = sum(k=0, 31, (n+k)^(3*k) * x^k/k! +O(x^31))
Vec(round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^200) )*1.) )
  • PARI
    {L(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)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 15 2021

Formula

E.g.f. exp( Sum_{n>=0} L(n)*x^n/n! ), where L(n) = [x^n*y^(2*n+1)/n!] log( Sum_{n>=0} (n + y)^(3*n) * x^n/n! ). - Paul D. Hanna, Jul 15 2021
a(n) ~ 3^(n + 1/2) * (3 + sqrt(6))^(n - 1/2) * exp((2-sqrt(6))*n - 2*sqrt(6) + 5) * n^(n-2) / 2^(n + 3/2). - Vaclav Kotesovec, Mar 20 2024

A266484 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 11, 316, 15741, 1140376, 109350271, 13100626176, 1886686497401, 317762099341696, 61318533545522451, 13343942849386863616, 3233753469962945660341, 863794149132594286734336, 252178372791563562485494151, 79890921514691257167186558976, 27298165065421976828646695794161, 10007689235634878438090676073824256, 3918413783588692571816707646546345371, 1631982989611299844119224469019967225856, 720447625733586591482575137323090206302701
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!.

Examples

			E.g.f.: 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! + 317762099341696*x^9/9! + 61318533545522451*x^10/10! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N+1)^5*(x/N^4) + (N+2)^10*(x/N^4)^2/2! + (N+3)^15*(x/N^4)^3/3! + (N+4)^20*(x/N^4)^4/4! + (N+5)^25*(x/N^4)^5/5! + (N+6)^30*(x/N^4)^6/6! +...]^(1/N).

Links

Crossrefs

Cf. A266481, A266482, A266483, A266485, A266486.

Programs

  • PARI
    /* Informal listing of terms 0..30 */
\p500
P(n) = sum(k=0,32, (n+k)^(5*k) * x^k/k! +O(x^32))
Vec(round(serlaplace( subst(P(10^100)^(1/10^100),x,x/10^400) )*1.) )
  • PARI
    /* Using logarithmic formula */
{L(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)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 29 2023

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.
(1) A(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} L(n)*x^n/n! ), where L(n) = [x^n*y^(4*n+1)/n!] log( Sum_{n>=0} (n + y)^(5*n) * x^n/n! ). - Paul D. Hanna, Jan 29 2023
a(n) ~ 5^(5*n/2 + 1/4) * (1 + sqrt(5))^(3*n - 3/2) * exp((4-2*sqrt(5))*n - 4*sqrt(5) + 9) * n^(n-2) / 2^(7*n + 1). - Vaclav Kotesovec, Mar 20 2024

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

A266487 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N - n)^(2*n) * (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, -3, 31, -559, 14541, -496811, 21081859, -1070585055, 63366015673, -4285932328819, 326248732427751, -27610580638457807, 2572239828612623365, -261621661000490429211, 28849626308051995285771, -3428690784657696770593471, 436924188109882619766249201, -59432725217403244945921112675, 8595527924368773785463788378287, -1317123285394547040368548520041839, 213171869078193696050387803319525821, -36338236299957647745418230431675850763, 6507698606647750492700809995200106342675, -1221579456277487714539848255959245396739999
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!.

Examples

			E.g.f.: A(x) = 1 + x - 3*x^2/2! + 31*x^3/3! - 559*x^4/4! + 14541*x^5/5! - 496811*x^6/6! + 21081859*x^7/7! - 1070585055*x^8/8! + 63366015673*x^9/9! - 4285932328819*x^10/10! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N-1)^2*(x/N) + (N-2)^4*(x/N)^2/2! + (N-3)^6*(x/N)^3/3! + (N-4)^8*(x/N)^4/4! + (N-5)^10*(x/N)^5/5! + (N-6)^12*(x/N)^6/6! +...]^(1/N).

Crossrefs

Cf. A266481, A266482, A266483, A266484, A266485, A266486.

Programs

  • PARI
    /* Informal listing of terms 0..30 */
\p300
H(n) = sum(k=0,32, (n - k)^(2*k) * x^k/k! +O(x^32))
Vec( round( serlaplace( subst(H(10^100)^(1/10^100),x,x/10^100) )*1.) )

A266488 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(2*n) * (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 0, 1, 4, 0, 1, 12, 42, 0, 1, 24, 216, 752, 0, 1, 40, 660, 5440, 19360, 0, 1, 60, 1560, 22320, 178920, 654912, 0, 1, 84, 3150, 68320, 916440, 7316064, 27546736, 0, 1, 112, 5712, 173600, 3432800, 44693376, 359051392, 1388207872, 0, 1, 144, 9576, 387072, 10493280, 197261568, 2536797312, 20605529088, 81621893376, 0, 1, 180, 15120, 782880, 27735120, 702777600, 12845683200, 164732083200, 1355581612800, 5488951731200, 0
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!.

Examples

			E.g.f. A(x) = 1 + x +
x^2/2! * (1 + 4*y) +
x^3/3! * (1 + 12*y + 42*y^2) +
x^4/4! * (1 + 24*y + 216*y^2 + 752*y^3) +
x^5/5! * (1 + 40*y + 660*y^2 + 5440*y^3 + 19360*y^4) +
x^6/6! * (1 + 60*y + 1560*y^2 + 22320*y^3 + 178920*y^4 + 654912*y^5) +
x^7/7! * (1 + 84*y + 3150*y^2 + 68320*y^3 + 916440*y^4 + 7316064*y^5 + 27546736*y^6) +
x^8/8! * (1 + 112*y + 5712*y^2 + 173600*y^3 + 3432800*y^4 + 44693376*y^5 + 359051392*y^6 + 1388207872*y^7) + ...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N + y)^2*(x/N) + (N + 2*y)^4*(x/N)^2/2! + (N + 3*y)^6*(x/N)^3/3! + (N + 4*y)^8*(x/N)^4/4! + (N + 5*y)^10*(x/N)^5/5! + (N + 6*y)^12*(x/N)^6/6! +...]^(1/N).
Triangle of coefficients T(n,k) of x^n*y^k/n!, n>=0, k=0..n, begins:
1;
1, 0;
1, 4, 0;
1, 12, 42, 0;
1, 24, 216, 752, 0;
1, 40, 660, 5440, 19360, 0;
1, 60, 1560, 22320, 178920, 654912, 0;
1, 84, 3150, 68320, 916440, 7316064, 27546736, 0;
1, 112, 5712, 173600, 3432800, 44693376, 359051392, 1388207872, 0;
1, 144, 9576, 387072, 10493280, 197261568, 2536797312, 20605529088, 81621893376, 0;
1, 180, 15120, 782880, 27735120, 702777600, 12845683200, 164732083200, 1355581612800, 5488951731200, 0;
1, 220, 22770, 1467840, 65659440, 2143842624, 52117998240, 938463651840, 12065358919680, 100649306644480, 415721105434624, 0; ...

Crossrefs

Cf. A266481, A266482, A266483, A266484, A266485, A266486, A266487.

Programs

  • PARI
    /* Print the initial rows of this triangle: */
\p400
P(n) = sum(k=0, 21, (n + k*y)^(2*k) * (x/n)^k/k! +O(x^21))
V=Vec( round( serlaplace( P(10^100)^(1/10^100) )*1.) )
for(n=1,15,print(Vec(V[n]+O(y^n))))
Showing 1-7 of 7 results.

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

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