A319147 -id:A319147 - cp's OEIS frontend

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

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

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!.
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).

			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! + ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

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

/* 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.) )

/* 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), ", "))

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

A319834 a(n) = coefficient of x^ny^(n+1)/n! in log( Sum_{n>=0} (n^2 + ny + y^2)^n * x^n/n! ).

Original entry on oeis.org

1, 2, 15, 184, 3325, 79056, 2345539, 83505920, 3472829721, 165321395200, 8868765212791, 529513463098368, 34831327847918485, 2503184803456354304, 195151614670701520875, 16405316791445973139456, 1479333355684885588136881, 142443466217414911148359680, 14587416733382035646737882591, 1583199811285962289889116160000

Offset: 1

Paul D. Hanna, Sep 30 2018

nonn

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! equals the logarithm of the e.g.f. of A319147.

			E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3325*x^5/5! + 79056*x^6/6! + 2345539*x^7/7! + 83505920*x^8/8! + 3472829721*x^9/9! + ...
Exponentiation yields the e.g.f. of A319147:
exp(A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4776*x^5/5! + 111967*x^6/6! + 3280264*x^7/7! + 115550073*x^8/8! +...+ A319147(n)*x^n/n! + ...
which equals
Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n + n^2)^n * (x/N)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 1, 5, 46, 665, 13176, 335077, 10438240, 385869969, 16532139520, ...].

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

Cf. A319147, A318634.

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

a(n) ~ c * d^n * n! / n^(5/2), where d = 6.1601834100761946... (same as for A319147) and c = 0.193396776821391327... - Vaclav Kotesovec, Mar 19 2024

A359927 E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + 3Nn + 2n^2)^n (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 7, 112, 2965, 111856, 5528419, 339433984, 24965493865, 2142654088960, 210377086601311, 23269631260880896, 2864038963868253373, 388330717110688399360, 57521524729462484086075, 9242821569458332441378816, 1601434996324769244061560529

Offset: 0

Paul D. Hanna, Jan 20 2023

nonn

Related limits:
(C1) exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
(C2) 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).

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 112*x^3/3! + 2965*x^4/4! + 111856*x^5/5! + 5528419*x^6/6! + 339433984*x^7/7! + 24965493865*x^8/8! + 2142654088960*x^9/9! + 210377086601311*x^10/10! + 23269631260880896*x^11/11! + 2864038963868253373*x^12/12! + ...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N^2+3*N+2)*(x/N) + (N^2+3*2*N+2*2^2)^2*(x/N)^2/2! + (N^2+3*3*N+2*3^2)^3*(x/N)^3/3! + (N^2+3*4*N+2*4^2)^4*(x/N)^4/4! + (N^2+3*5*N+2*5^2)^5*(x/N)^5/5! + (N^2+3*6*N+2*6^2)^6*(x/N)^6/6! + ...]^(1/N).
RELATED SERIES.
The logarithm of the g.f. A(x) begins:
(a) log(A(x)) = x + 6*x^2/2! + 93*x^3/3! + 2448*x^4/4! + 92505*x^5/5! + 4589568*x^6/6! + 283008621*x^7/7! + 20903023872*x^8/8! + ... + A359928(n)*x^n/n! + ...
where A359928(n) = [x^n*y^(n+1)/n!] (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 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!, n >= 1, in the series given by
(b) (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 2*y^2)^n * x^n/n! ) = x*(y^2 + 3/2*y + 1/2) + x^2/2!*(6*y^3 + 39/2*y^2 + 21*y + 15/2) + x^3/3!*(93*y^4 + 999/2*y^3 + 2055/2*y^2 + 1917/2*y + 683/2) + x^4/4!*(2448*y^5 + 19119*y^4 + 61704*y^3 + 102742*y^2 + 88080*y + 31019) + x^5/5!*(92505*y^6 + 1948347/2*y^5 + 8887325/2*y^4 + 11224575*y^3 + 16525750*y^2 + 26820135/2*y + 9342629/2) + x^6/6!*(4589568*y^7 + 61994772*y^6 + 374546664*y^5 + 1310466240*y^4 + 2862046080*y^3 + 3891543876*y^2 + 3041064504*y + 1050241608) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A359928, A319147, A266481, A318633.

/* Using formula for the logarithm of g.f. A(x) */
{L(n) = (1/2) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^m*(m + 2*y)^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), ", "))

/* Using limit formula */
\p100
P(n) = sum(k=0, 31, ((n + k)*(n + 2*k))^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. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.
(1) A(x) = lim_{N->oo} [ Sum_{n>=0} (N + n)^n*(N + 2*n)^n * (x/N)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} A359928(n)*x^n/n! ), where A359928(n) = (1/2) * [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n + y)^n*(n + 2*y)^n *x^n/n! ).
a(n) ~ c * n! * d^n / n^(5/2), where d = 12.7029497597456784744445675253711147535742245945208995646... and c = 0.17380315134029681101563539591890111670852050181568... - Vaclav Kotesovec, Mar 14 2023

A359928 a(n) = coefficient of x^ny^(n+1)/n! in (1/2) log( Sum_{n>=0} (n^2 + 3ny + 2y^2)^n x^n/n! ).

Original entry on oeis.org

1, 6, 93, 2448, 92505, 4589568, 283008621, 20903023872, 1800986581521, 177455695795200, 19690717755237309, 2430478269127673856, 330392930155527272553, 49053029845102480576512, 7898602773992589665290125, 1371137549213022697047785472, 255275516636592894833768588961

Offset: 1

Paul D. Hanna, Jan 20 2023

nonn

			E.g.f.: A(x) = x + 6*x^2/2! + 93*x^3/3! + 2448*x^4/4! + 92505*x^5/5! + 4589568*x^6/6! + 283008621*x^7/7! + 20903023872*x^8/8! + 1800986581521*x^9/9! + 177455695795200*x^10/10! + ...
Exponentiation yields the e.g.f. of A319147:
exp(A(x)) = 1 + x + 7*x^2/2! + 112*x^3/3! + 2965*x^4/4! + 111856*x^5/5! + 5528419*x^6/6! + 339433984*x^7/7! + 24965493865*x^8/8! + 2142654088960*x^9/9! + ... + A319147(n)*x^n/n! + ...
which equals the limit
exp(A(x)) = lim_{N->oo} [ Sum_{n>=0} (N^2 + 3*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, 3, 31, 612, 18501, 764928, 40429803, 2612877984, 200109620169, ...].

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

Cf. A359927, A319834, A318634.

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

E.g.f. A(x) = Sum_{n>=1} a(n) * x^n/n! may be defined by the following.
(1) a(n) = [x^n*y^(n+1)/n!] (1/2)*log( Sum_{n>=0} (n + y)^n*(n + 2*y)^n *x^n/n! ).
(2) A(x) = lim_{N->oo} (1/N)*log( Sum_{n>=0} (N + n)^n*(N + 2*n)^n * (x/N)^n/n! ).
a(n) ~ c * d^n * n! / n^(5/2), where d = 12.7029497597456784744445675253711147535742245945208995646083627... and c = 0.15440395598650604464793307483290467035754174771895993579108... - Vaclav Kotesovec, Mar 21 2024

A359917 E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + Nn + 2n^2)^n * (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 3, 28, 413, 9216, 268327, 9831424, 432251577, 22259307520, 1313366140331, 87431498993664, 6482838033725077, 529958491541291008, 47356678577690489295, 4592761099982656823296, 480465410003489098874993, 53933291626260492656050176, 6466413087139041540884403667

Offset: 0

Paul D. Hanna, Jan 21 2023

nonn

Related limits:
(C1) exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
(C2) 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).

			E.g.f.: 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! + 22259307520*x^9/9! + 1313366140331*x^10/10! + ...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N^2+N+2)*(x/N) + (N^2+2*N+2*2^2)^2*(x/N)^2/2! + (N^2+3*N+2*3^2)^3*(x/N)^3/3! + (N^2+4*N+2*4^2)^4*(x/N)^4/4! + (N^2+5*N+2*5^2)^5*(x/N)^5/5! + (N^2+6*N+2*6^2)^6*(x/N)^6/6! + ...]^(1/N).
RELATED SERIES.
The logarithm of the g.f. A(x) begins:
(a) log(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! + ... + A359918(n)*x^n/n! + ...
where A359918(n) = [x^n*y^(n+1)/n!] (1/2) * log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! );
that is, the coefficients in the logarithm of e.g.f A(x) equals the coefficients of y^(n+1)*x^n/n! in the series given by
(b) (1/2) * log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! ) = (y^2 + 1/2*y + 1/2)*x + (2*y^3 + 15/2*y^2 + 7*y + 15/2)*x^2/2! + (21*y^4 + 197/2*y^3 + 543/2*y^2 + 639/2*y + 683/2)*x^3/3! + (304*y^5 + 2495*y^4 + 8984*y^3 + 22246*y^2 + 29360*y + 31019)*x^4/4! + (6985*y^6 + 150489/2*y^5 + 817005/2*y^4 + 1335885*y^3 + 3162830*y^2 + 8940045/2*y + 9342629/2)*x^5/5! + (205056*y^7 + 2946228*y^6 + 20587128*y^5 + 94146240*y^4 + 294518400*y^3 + 684700836*y^2 + 1013688168*y + 1050241608)*x^6/6! + ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A359918, A266485, A359927, A319147, A266481, A318633, A319834.

/* Using formula for the logarithm of g.f. A(x) */
{L(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)}
{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), ", "))

/* Using limit formula */
\p100
P(n) = sum(k=0, 31, (n^2 + n*k + 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.) )

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.
(1) A(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} A359918(n)*x^n/n! ), where A359918(n) = (1/2) * [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! ).
a(n) ~ c * d^n * n! / n^(5/2), where d = 7.68892218919697462312... and c = 0.155267010681833... - Vaclav Kotesovec, Mar 21 2024

cp's OEIS Frontend

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

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A319834 a(n) = coefficient of x^n*y^(n+1)/n! in log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! ).

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A359927 E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + 3*N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).

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A359928 a(n) = coefficient of x^n*y^(n+1)/n! in (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 2*y^2)^n * x^n/n! ).

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A359917 E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).

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

A319834 a(n) = coefficient of x^ny^(n+1)/n! in log( Sum_{n>=0} (n^2 + ny + y^2)^n * x^n/n! ).

A359927 E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + 3Nn + 2n^2)^n (x/N)^n/n! ]^(1/N).

A359928 a(n) = coefficient of x^ny^(n+1)/n! in (1/2) log( Sum_{n>=0} (n^2 + 3ny + 2y^2)^n x^n/n! ).

A359917 E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + Nn + 2n^2)^n * (x/N)^n/n! ]^(1/N).