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 1-3 of 3 results.

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! ).

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Jan 20 2023

Keywords

Examples

			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, ...].

Links

Crossrefs

Cf. A266485, A359928, A319834, A318634.

Programs

  • PARI
    {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), ", "))

Formula

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

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).

Original entry on oeis.org

1, 1, 7, 112, 2965, 111856, 5528419, 339433984, 24965493865, 2142654088960, 210377086601311, 23269631260880896, 2864038963868253373, 388330717110688399360, 57521524729462484086075, 9242821569458332441378816, 1601434996324769244061560529
Offset: 0

Views

Author

Paul D. Hanna, Jan 20 2023

Keywords

Comments

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).

Examples

			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) + ...

Links

Crossrefs

Cf. A359928, A319147, A266481, A318633.

Programs

  • PARI
    /* 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), ", "))
  • PARI
    /* 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.) )

Formula

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

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 1-3 of 3 results.

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