A266526 -id:A266526 - cp's OEIS frontend

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

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

Paul D. Hanna, Dec 30 2015

nonn

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.

			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.

Paul D. Hanna, Table of n, a(n) for n = 0..100

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

{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),", "))

/* 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.) )

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

A318634 a(n) = coefficient of x^(2n-1)y^(2n)/(2n-1)! in Log( Sum_{n>=0} (n^2 + y^2)^n * x^n/n! ), for n>=1.

Original entry on oeis.org

1, 6, 480, 122640, 66044160, 61482516480, 88135315107840, 180378921026304000, 499734635092800307200, 1801642618822079338905600, 8199046303785011864744755200, 45976521975711536997953490124800, 311502479360401852390993821696000000, 2508845886467091418046335123571343360000, 23693183471722887844366765687378500648960000

Offset: 1

Paul D. Hanna, Sep 04 2018

nonn

E.g.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)! equals the logarithm of the e.g.f. of A318633.

			E.g.f.: A(x) = x + 6*x^3/3! + 480*x^5/5! + 122640*x^7/7! + 66044160*x^9/9! + 61482516480*x^11/11! + 88135315107840*x^13/13! + 180378921026304000*x^15/15! + ...
The e.g.f. A(x) may also be written using somewhat reduced coefficients
A(x) = x + x^3 + 8*x^5/2! + 146*x^7/3! + 4368*x^9/4! + 184832*x^11/5! + 10190656*x^13/6! + 695211120*x^15/7! + 56648897024*x^17/8! + 5374487515904*x^19/9! + ... + a(n)*(n-1)!/(2*n-1)! * x^(2*n-1)/(n-1)! + ...
Exponentiation yields the e.g.f. of A318633:
exp(A(x)) = 1 + x + x^2/2! + 7*x^3/3! + 25*x^4/4! + 541*x^5/5! + 3361*x^6/6! + 135451*x^7/7! + 1179697*x^8/8! + 72062425*x^9/9! +...+ A318633(n)*x^n/n! + ...
which equals
Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ]^(1/N).

Vaclav Kotesovec, Table of n, a(n) for n = 1..145 (terms 1..75 from Paul D. Hanna)

Cf. A318633, A266526, A319834.

{a(n) = (2*n-1)! * polcoeff( polcoeff( log( sum(m=0, 2*n, (m^2 + y^2)^m *x^m/m! ) +x*O(x^(2*n)) ), 2*n-1, x), 2*n, y)}
for(n=1, 20, print1(a(n), ", "))

a(n) ~ 5^(-1/4) * 2^(3*n - 3/2) * (1 + sqrt(5))^(n - 3/2) * exp((1 - sqrt(5))*n + (sqrt(5) - 3)/2) * n^(2*n-3). - Vaclav Kotesovec, Mar 20 2024

A266521 E.g.f.: Log( Sum_{n>=0} (n + y)^(2n) x^n/n! ) = Sum_{n>=1} Sum_{k=0..n+1} T(n,k) * x^n*y^k/n!, as a triangle of coefficients T(n,k) read by rows.

Original entry on oeis.org

1, 2, 1, 15, 28, 18, 4, 683, 1278, 933, 316, 42, 62038, 117440, 92680, 38240, 8272, 752, 9342629, 17880090, 14855385, 6881640, 1880340, 288048, 19360, 2100483216, 4054752672, 3490688496, 1743156480, 547098240, 108228192, 12523584, 654912, 658746323647, 1279910119670, 1130161189549, 594323331364, 204256939502, 47125635760, 7147508032, 652959872, 27546736, 274730459045232, 536368375356928, 482514140459520, 263340552849920, 96404466197760, 24628940050176, 4404380994048, 533057051648, 39701769216, 1388207872

Offset: 1

Paul D. Hanna, Jan 01 2016

Row sums form A266520, coefficients in Log( Sum_{n>=0} (n+1)^(2*n) * x^n/n! ).
Column 0 forms A266519, coefficients in log( Sum_{n>=0} n^(2*n) * x^n/n! ).
Rightmost border is A266526.

			E.g.f.: A(x,y) = x * (1 + 2*y + y^2) +
x^2/2! * (15 + 28*y + 18*y^2 + 4*y^3) +
x^3/3! * (683 + 1278*y + 933*y^2 + 316*y^3 + 42*y^4) +
x^4/4! * (62038 + 117440*y + 92680*y^2 + 38240*y^3 + 8272*y^4 + 752*y^5) +
x^5/5! * (9342629 + 17880090*y + 14855385*y^2 + 6881640*y^3 + 1880340*y^4 + 288048*y^5 + 19360*y^6) +
x^6/6! * (2100483216 + 4054752672*y + 3490688496*y^2 + 1743156480*y^3 + 547098240*y^4 + 108228192*y^5 + 12523584*y^6 + 654912*y^7) +...
where
exp(A(x,y)) = 1 + (1 + y)*x + (2 + y)^4*x^2/2! + (3 + y)^6*x^3/3! + (4 + y)^8*x^4/4! + (5 + y)^10*x^5/5! + (6 + y)^12*x^6/6! +...
This triangle begins:
1, 2, 1;
15, 28, 18, 4;
683, 1278, 933, 316, 42;
62038, 117440, 92680, 38240, 8272, 752;
9342629, 17880090, 14855385, 6881640, 1880340, 288048, 19360;
2100483216, 4054752672, 3490688496, 1743156480, 547098240, 108228192, 12523584, 654912;
658746323647, 1279910119670, 1130161189549, 594323331364, 204256939502, 47125635760, 7147508032, 652959872, 27546736;
274730459045232, 536368375356928, 482514140459520, 263340552849920, 96404466197760, 24628940050176, 4404380994048, 533057051648, 39701769216, 1388207872;
147034646085347145, 288100398039817266, 262835789583073329, 147457696629622032, 56514667400140392, 15510808994500512, 3097157140510272, 445604738641920, 44324678623680, 2758053332736, 81621893376; ...

Paul D. Hanna, Table of n, a(n) for n = 1..1125 as a flattened triangle read by rows 1..45.

Cf. A266519, A266520, A266526.

{T(n,k) = n! * polcoeff( polcoeff( log( sum(m=0,n+1, (m + y)^(2*m) *x^m/m! ) +x*O(x^n) ),n,x), k,y)}
for(n=1,10, for(k=0,n+1, print1(T(n,k),", "));print(""))

A360238 a(n) = [y^nx^n/n] log( Sum_{m>=0} (m + y)^(2m) * x^m ) for n >= 1.

Original entry on oeis.org

2, 42, 1376, 60934, 3377252, 224036904, 17282039280, 1519096411230, 149867251224092, 16398595767212452, 1971137737765484444, 258215735255164847944, 36617351885600586385222, 5588967440618883091216208, 913592455995572681826313856, 159241707066923571547572653630

Offset: 1

Paul D. Hanna, Feb 11 2023

nonn

Related sequence: A000984(n) = binomial(2*n,n) = [y^n*x^n/n] log( Sum_{m>=0} (1 + y)^(2*m) * x^m ) for n >= 1.

			L.g.f.: A(x) = 2*x + 42*x^2/2 + 1376*x^3/3 + 60934*x^4/4 + 3377252*x^5/5 + 224036904*x^6/6 + 17282039280*x^7/7 + 1519096411230*x^8/8 + ...
a(n) equals the coefficient of y^n*x^n/n in the logarithmic series:
log( Sum_{m>=0} (m + y)^(2*m) * x^m ) = (y^2 + 2*y + 1)*x + (y^4 + 12*y^3 + 42*y^2 + 60*y + 31)*x^2/2 + (y^6 + 30*y^5 + 297*y^4 + 1376*y^3 + 3348*y^2 + 4188*y + 2140)*x^3/3 + (y^8 + 56*y^7 + 1100*y^6 + 10792*y^5 + 60934*y^4 + 209464*y^3 + 436692*y^2 + 510952*y + 258779)*x^4/4 + (y^10 + 90*y^9 + 2945*y^8 + 49960*y^7 + 510160*y^6 + 3377252*y^5 + 14971780*y^4 + 44457000*y^3 + 85336175*y^2 + 96141170*y + 48446971)*x^5/5 + (y^12 + 132*y^11 + 6486*y^10 + 169236*y^9 + 2730921*y^8 + 29547696*y^7 + 224036904*y^6 + 1214958240*y^5 + 4717830978*y^4 + 12868488144*y^3 + 23497266672*y^2 + 25858665696*y + 12994749280)*x^6/6 + ...
Exponentiation yields the g.f. of A360239:
exp(A(x)) = 1 + 2*x + 23*x^2 + 502*x^3 + 16414*x^4 + 716936*x^5 + 39167817*x^6 + 2567058766*x^7 + 196159319943*x^8 + ... + A360239(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A360239, A360348, A266526, A000984, A059304, A098658.

{a(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(2*m) *x^m ) +x*O(x^n) ), n, x), n, y)}
for(n=0,20,print1(a(n),", "))

a(n) ~ (1 - exp(-1)/4) * 2^(2*n) * n^(n + 1/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 12 2023

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

Paul D. Hanna, Feb 10 2023

nonn

			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, ...].

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A266526, A360339, A360341, A359926.

/* 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), ", "))

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

cp's OEIS Frontend

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

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A318634 a(n) = coefficient of x^(2*n-1)*y^(2*n)/(2*n-1)! in Log( Sum_{n>=0} (n^2 + y^2)^n * x^n/n! ), for n>=1.

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A266521 E.g.f.: Log( Sum_{n>=0} (n + y)^(2*n) * x^n/n! ) = Sum_{n>=1} Sum_{k=0..n+1} T(n,k) * x^n*y^k/n!, as a triangle of coefficients T(n,k) read by rows.

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A360238 a(n) = [y^n*x^n/n] log( Sum_{m>=0} (m + y)^(2*m) * x^m ) for n >= 1.

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

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

A318634 a(n) = coefficient of x^(2n-1)y^(2n)/(2n-1)! in Log( Sum_{n>=0} (n^2 + y^2)^n * x^n/n! ), for n>=1.

A266521 E.g.f.: Log( Sum_{n>=0} (n + y)^(2n) x^n/n! ) = Sum_{n>=1} Sum_{k=0..n+1} T(n,k) * x^n*y^k/n!, as a triangle of coefficients T(n,k) read by rows.

A360238 a(n) = [y^nx^n/n] log( Sum_{m>=0} (m + y)^(2m) * x^m ) for n >= 1.

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