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.

A386655 E.g.f.: Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.

Original entry on oeis.org

1, 3, 23, 708, 82677, 39043808, 75384175459, 594418947869568, 19030890530555146281, 2460681168464503636482816, 1280084112577610436036966382815, 2672769582069469500760580570122074560, 22366167041278673568399013569832022272725469
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2025

Keywords

Comments

Conjecture: for n >= 6, a(n) (mod 6) equals [1, 0, 3, 0, 3, 2] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = 2 with p = LambertW(x)/x, r = x.

Examples

			E.g.f.: A(x) = 1 + 3*x + 23*x^2/2! + 708*x^3/3! + 82677*x^4/4! + 39043808*x^5/5! + 75384175459*x^6/6! + 594418947869568*x^7/7! + ...
where A(x) = Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.

Crossrefs

Cf. A386656 (q=3), A386657 (q=4), A386658 (q=5), A386648.

Programs

  • Mathematica
    nmax = 15; Join[{1}, Rest[CoefficientList[Series[Sum[(2^k*x + LambertW[x])^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]] (* Vaclav Kotesovec, Aug 23 2025 *)
  • PARI
    {a(n) = sum(k=0,n, binomial(n,k) * 2^(k*(k+1)) * (2^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))
  • PARI
    {a(n) = my(A = sum(m=0, n, (2^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 2^(n^2) * exp( LambertW(x) * 2^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 2^(n^2) * (x/LambertW(x))^(2^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 2^(n*(n+1)) * x^n/n! * Sum_{k>=0} (2^n - k)^(k-1) * x^k/k!.
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(k*(k+1)) * (2^k - (n-k))^(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n*k) * (1 - (n-k)/2^k)^(n-k-1).
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Aug 23 2025

A386656 E.g.f.: Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.

Original entry on oeis.org

1, 4, 98, 21901, 45203076, 864855654349, 151334120052647134, 240066304912259832915171, 3437872829353908000927273009224, 443629285010311848968435132228644809721, 515464807017361539745514781011221080738833641050
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2025

Keywords

Comments

Conjecture: for n >= 6, a(n) (mod 6) equals [4, 3, 2, 1, 0, 1] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = 3 with p = LambertW(x)/x, r = x.

Examples

			E.g.f.: A(x) = 1 + 4*x + 98*x^2/2! + 21901*x^3/3! + 45203076*x^4/4! + 864855654349*x^5/5! + 151334120052647134*x^6/6! + ...
where A(x) = Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.

Crossrefs

Cf. A386655 (q=2), A386657 (q=4), A386658 (q=5), A386648.

Programs

  • PARI
    {a(n) = sum(k=0,n, binomial(n,k) * 3^(k*(k+1)) * (3^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))
  • PARI
    {a(n) = my(A = sum(m=0, n, (3^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 3^(n^2) * exp( LambertW(x) * 3^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 3^(n^2) * (x/LambertW(x))^(3^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 3^(n*(n+1)) * x^n/n! * Sum_{k>=0} (3^n - k)^(k-1) * x^k/k!.
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(k*(k+1)) * (3^k - (n-k))^(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n*k) * (1 - (n-k)/3^k)^(n-k-1).

A386657 E.g.f.: Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.

Original entry on oeis.org

1, 5, 287, 274532, 4362420261, 1131407873777920, 4729288202285254702123, 317048074495318899943286044736, 340323907513179399929311813628104334217, 5846207259092593125133941613189798019292422881280
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2025

Keywords

Comments

Conjecture: for n >= 6, a(n) (mod 6) equals [1, 2, 3, 2, 3, 4] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = 4 with p = LambertW(x)/x, r = x.

Examples

			E.g.f.: A(x) = 1 + 5*x + 287*x^2/2! + 274532*x^3/3! + 4362420261*x^4/4! + 1131407873777920*x^5/5! + 4729288202285254702123*x^6/6! + ...
where A(x) = Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.

Crossrefs

Cf. A386655 (q=2), A386656 (q=3), A386658 (q=5), A386648.

Programs

  • PARI
    {a(n) = sum(k=0,n, binomial(n,k) * 4^(k*(k+1)) * (4^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))
  • PARI
    {a(n) = my(A = sum(m=0, n, (4^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 4^(n^2) * exp( LambertW(x) * 4^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 4^(n^2) * (x/LambertW(x))^(4^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 4^(n*(n+1)) * x^n/n! * Sum_{k>=0} (4^n - k)^(k-1) * x^k/k!.
a(n) = Sum_{k=0..n} binomial(n,k) * 4^(k*(k+1)) * (4^k - (n-k))^(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * 4^(n*k) * (1 - (n-k)/4^k)^(n-k-1).
Showing 1-3 of 3 results.

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