A318634 -id:A318634 - cp's OEIS frontend

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

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

A318633 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, 1, 7, 25, 541, 3361, 135451, 1179697, 72062425, 800549281, 66556219951, 895033831561, 94883671398517, 1494100965599425, 193406329375101571, 3484870135007879521, 534176530923146533681, 10828212525851495841217, 1921147013664776094354775, 43243444110218718242048761, 8725818848745836121239731021, 215849236633217233817102676961, 48851798675231712377778383815147

Offset: 0

Paul D. Hanna, Sep 04 2018

nonn

Compare to:
(C1) exp(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
(C2) W(x) = Limit_{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 + 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! + 800549281*x^10/10! + ...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N^2+1)*(x/N) + (N^2+2^2)^2*(x/N)^2/2! + (N^2+3^2)^3*(x/N)^3/3! + (N^2+4^2)^4*(x/N)^4/4! + (N^2+5^2)^5*(x/N)^5/5! + (N^2+6^2)^6*(x/N)^6/6! +...]^(1/N).
RELATED SERIES.
(a) The following limit exists:
G(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ] / A(x)^N
where G(x) is an even function that begins
G(x) = 1 + 6*x^2/2! + 636*x^4/4! + 200520*x^6/6! + 127426320*x^8/8! + 135927439200*x^10/10! + 218711489895360*x^12/12! + 494824278347118720*x^14/14! 1497827553156635500800*x^16/16! +...
(b) The logarithm of the e.g.f. A(x) is an odd function that begins:
Log(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! + ... + A318634(n)*x^(2*n-1)/(2*n-1)! +...

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A318634, A266481.

{A318634(n) = 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, A318634(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^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. exp( Sum_{n>=0} A318634(n)*x^n/n! ), where A318634(n) = [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + y^2)^n *x^n/n! ).

a(n) ~ c * 2^(3*n/2) * (1 + sqrt(5))^(n/2) * exp((3 - sqrt(5))*n/2) * n^(n-2) / (2^(n-2) * exp(n)), where c = 0.06299092342072472381892574307... if n is even and c = 0.21603924353200238368086490887... if n is odd. - Vaclav Kotesovec, Mar 21 2024

A359926 a(n) = coefficient of x^ny^(n+1)/n! in (1/4) log( Sum_{n>=0} (n + 2y)^(2n) * x^n/n! ).

Original entry on oeis.org

1, 8, 168, 6016, 309760, 20957184, 1762991104, 177690607616, 20895204704256, 2810343286374400, 425698411965054976, 71735043897868419072, 13313460758336789020672, 2698754565131159025483776, 593332971403056575938560000, 140634107346363806457259884544

Offset: 1

Paul D. Hanna, Jan 20 2023

nonn

			E.g.f. 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! + 20895204704256*x^9/9! + 2810343286374400*x^10/10! + ...
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! where a(n) equals the coefficient of y^(n+1)*x^n/n! in the series given by
(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! + (309760*y^6 + 2304384*y^5 + 7521360*y^4 + 13763280*y^3 + 14855385*y^2 + 8940045*y + 9342629/4)*x^5/5! + (20957184*y^7 + 200377344*y^6 + 865825536*y^5 + 2188392960*y^4 + 3486312960*y^3 + 3490688496*y^2 + 2027376336*y + 525120804)*x^6/6! + ...
Exponentiation yields the e.g.f. of A266485:
exp(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! + ... + A266485(n)*x^n/n! + ...
which equals
lim_{N->oo} [ Sum_{n>=0} (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, 4, 56, 1504, 61952, 3492864, 251855872, 22211325952, 2321689411584, ...].

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

Cf. A266485, A359928, A319834, A318634.

{a(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)}
for(n=1, 30, print1(a(n), ", "))

a(n) ~ c * d^n * n! / n^(5/2), where d = 4*(1 + sqrt(2)) * exp(2 - sqrt(2)) = 17.347603772617734513447467379678826546908822081006190652539615... and c = sqrt((2 - sqrt(2))/Pi)/4 = 0.107953003168342979028946547859477378793474... - Vaclav Kotesovec, Feb 13 2023, updated Mar 17 2024

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

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

Paul D. Hanna, Jan 21 2023

nonn

			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, ...].

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

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

{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), ", "))

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

cp's OEIS Frontend

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! ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A318633 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ]^(1/N).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A359926 a(n) = coefficient of x^n*y^(n+1)/n! in (1/4) * log( Sum_{n>=0} (n + 2*y)^(2*n) * x^n/n! ).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

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! ).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

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! ).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

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

A359926 a(n) = coefficient of x^ny^(n+1)/n! in (1/4) log( Sum_{n>=0} (n + 2y)^(2n) * x^n/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! ).

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