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.

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A362776 E.g.f. satisfies A(x) = exp( x/(1-x)^2 * A(x)^2 ).

Original entry on oeis.org

1, 1, 9, 127, 2601, 70981, 2433673, 100697787, 4886085137, 272168650441, 17121437245161, 1200717094233559, 92892754255837561, 7859587210132504653, 721996671783802854377, 71564871858940414914451, 7613407794191946986893857, 865285095267929315207801233
Offset: 0

Views

Author

Seiichi Manyama, May 02 2023

Keywords

Links

Crossrefs

Cf. A082579, A362775.
Cf. A361065.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x/(1-x)^2)/2)))

Formula

E.g.f.: exp( -LambertW(-2*x/(1-x)^2)/2 ).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n+k-1,n-k)/k!.
From Vaclav Kotesovec, Nov 10 2023: (Start)
E.g.f.: (1-x) * sqrt(-LambertW(-2*x/(1-x)^2) / (2*x)).
a(n) ~ sqrt(-sqrt(1 + 2*exp(-1)) + 1 + 2*exp(-1)) * n^(n-1) / (sqrt(2) * (-1 + sqrt(1 + 2*exp(-1)))^(3/2) * (-sqrt(1 + 2*exp(-1)) + 1 + exp(-1))^(n - 1/2) * exp(2*n - 1/2)). (End)

A384819 Nonnegative numbers a(n) < n for n >= 1 such that exp( Sum_{n>=1} (n^2 - a(n))*x^n/n ) is a power series with integral coefficients.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 6, 1, 2, 7, 10, 3, 12, 11, 3, 1, 16, 3, 18, 15, 14, 19, 22, 3, 4, 23, 2, 27, 28, 12, 30, 1, 15, 31, 7, 3, 36, 35, 32, 7, 40, 37, 42, 7, 21, 43, 46, 3, 6, 7, 27, 11, 52, 3, 34, 35, 50, 55, 58, 44, 60, 59, 5, 1, 18, 0, 66, 19, 39, 40, 70, 3, 72, 71, 3, 23, 1, 19, 78, 55, 2, 79, 82, 41, 47, 83, 51, 47, 88, 84, 74, 31, 86, 91, 17, 3, 96, 11, 42, 15, 100
Offset: 1

Views

Author

Paul D. Hanna, Jun 18 2025

Keywords

Comments

Conjecture: a(p^n) = p - 1 when p is prime for n >= 1.

Examples

			L.g.f.: A(x) = 0*x + 1*x^2/2 + 2*x^3/3 + 1*x^4/4 + 4*x^5/5 + 3*x^6/6 + 6*x^7/7 + 1*x^8/8 + 2*x^9/9 + 7*x^10/10 + 10*x^11/11 + 3*x^12/12 + 12*x^13/13 + 11*x^14/14 + 3*x^15/15 + 1*x^16/16 + ...
where the following is a power series with integral coefficients
exp( x/(1-x)^2 - A(x) ) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 14*x^5 + 25*x^6 + 43*x^7 + 74*x^8 + 124*x^9 + 205*x^10 + 335*x^11 + 543*x^12 + 869*x^13 + 1379*x^14 + 2170*x^15 + 3388*x^16 + ... + A384820(n)*x^n + ...
which is equivalent to
exp( Sum_{n>=1} (n^2 - a(n))*x^n/n ) = exp(x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 21*x^5/5 + 33*x^6/6 + 43*x^7/7 + 63*x^8/8 + 79*x^9/9 + 93*x^10/10 + 111*x^11/11 + 141*x^12/12 + 157*x^13/13 + 185*x^14/14 + 222*x^15/15 + 255*x^16/16 + ...).

Links

Crossrefs

Cf. A384820, A082579 (exp(x/(1-x)^2)).

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=2,n, A = concat(A,t);
for(t=1,(#A)^2+1, if( denominator( eval(polcoef( exp( intformal(Ser(A)) ),#A)) )==1, A[#A] = t + (#A)*(#A-1); break)) ); n^2 - A[n]}
for(n=1,101, print1(a(n),", "))

A384820 G.f. A(x) = exp( Sum_{n>=1} (n^2 - A384819(n))*x^n/n ) where A384819(k) < k for k >= 1 such that A(x) is a power series with integral coefficients.

Original entry on oeis.org

1, 1, 2, 4, 8, 14, 25, 43, 74, 124, 205, 335, 543, 869, 1379, 2170, 3388, 5249, 8079, 12353, 18776, 28375, 42651, 63782, 94923, 140614, 207384, 304578, 445528, 649200, 942495, 1363447, 1965697, 2824676, 4046190, 5778273, 8227533, 11681632, 16540183, 23357053, 32898242
Offset: 0

Views

Author

Paul D. Hanna, Jun 18 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 14*x^5 + 25*x^6 + 43*x^7 + 74*x^8 + 124*x^9 + 205*x^10 + 335*x^11 + 543*x^12 + 869*x^13 + 1379*x^14 + 2170*x^15 + 3388*x^16 + ...
where log(A(x)) = x/(1-x)^2 - D(x) and D(x) is the l.g.f. of A384819:
D(x) = 0*x + 1*x^2/2 + 2*x^3/3 + 1*x^4/4 + 4*x^5/5 + 3*x^6/6 + 6*x^7/7 + 1*x^8/8 + 2*x^9/9 + 7*x^10/10 + 10*x^11/11 + 3*x^12/12 + 12*x^13/13 + 11*x^14/14 + 3*x^15/15 + 1*x^16/16 + ... + A384819(n)*x^n/n + ...

Links

Crossrefs

Cf. A384819, A082579 (exp(x/(1-x)^2)).

Programs

  • PARI
    {a(n) = my(L=[1],A=1); for(i=1,n, L = concat(L,t);
for(t=1,(#L)^2+1, if( denominator( eval(polcoef( A = exp( intformal(Ser(L)) ),#L)) )==1, L[#L] = t + (#L)*(#L-1); break)) ); polcoef(A,n)}
for(n=0,40, print1(a(n),", "))

A386514 Expansion of e.g.f. exp(x^2/(1-x)^3).

Original entry on oeis.org

1, 0, 2, 18, 156, 1560, 18480, 254520, 3973200, 68947200, 1312748640, 27175024800, 607314818880, 14566195163520, 373027570755840, 10154293067318400, 292659790712889600, 8899747730037964800, 284685195814757337600, 9553060139009702515200, 335468448755976164428800
Offset: 0

Views

Author

Enrique Navarrete, Aug 23 2025

Keywords

Comments

For n > 0, a(n) is the number of ways to linearly order n distinguishable objects into one or several lines and then choose 2 objects from each line. If the lines are also linearly ordered see A364524.
A001804(n) is the number of ways if only 1 line is used.

Examples

			a(6)=18480 since there are 10800 ways using one line, 4320 ways with 2 lines using 2 and 4 objects, 3240 ways with 2 lines of 3 objects each, and 120 ways with 3 lines of 2 objects each.

Crossrefs

Cf. A001804, A364524.
Cf. A293012, A000262, A082579, A091695, A361283.
Cf. A052845, A052887, A387244.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[E^(x^2/(1-x)^3), {x, 0, nmax}], x] * Range[0, nmax]! (* or *)
nmax = 20; Join[{1}, Table[n!*Sum[Binomial[n + k - 1, 3*k - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* Vaclav Kotesovec, Aug 24 2025 *)

Formula

From Vaclav Kotesovec, Aug 24 2025: (Start)
For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n+k-1, 3*k-1) / k!.
a(n) = 4*(n-1)*a(n-1) - 2*(n-1)*(3*n-7)*a(n-2) + (n-2)*(n-1)*(4*n-11)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(1/8) * exp(1/27 - 3^(-5/4)*n^(1/4)/8 - 3^(-1/2)*n^(1/2)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n-1/8) / 2. (End)

A308417 Expansion of e.g.f. exp(x*(1 + x + x^2)/(1 - x^2)^2).

Original entry on oeis.org

1, 1, 3, 25, 145, 1461, 14011, 169933, 2231265, 32572585, 528302611, 9146070561, 174016032433, 3498446485405, 75954922790475, 1737982233878101, 42327522277348801, 1084073452000879953, 29253450397798616995, 827617575903336189865, 24503022168956714812881
Offset: 0

Views

Author

Ilya Gutkovskiy, May 25 2019

Keywords

Crossrefs

Cf. A007434, A026741, A082579, A088009, A301876, A308418.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[x (1 + x + x^2)/(1 - x^2)^2], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Product[(1 + x^k)^(DirichletConvolve[j^2, MoebiusMu[j], j, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Numerator[k/2] k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]
  • PARI
    my(x ='x + O('x^30)); Vec(serlaplace(exp(x*(1 + x + x^2)/(1 - x^2)^2))) \\ Michel Marcus, May 26 2019

Formula

E.g.f.: exp(Sum_{k>=1} A026741(k)*x^k).
E.g.f.: Product_{k>=1} (1 + x^k)^(J_2(k)/k), where J_2() is the Jordan function (A007434).
a(0) = 1; a(n) = Sum_{k=1..n} A026741(k)*k!*binomial(n-1,k-1)*a(n-k).
a(n) ~ 2^(-1/6) * 3^(-1/3) * n^(n - 1/6) * exp((3/2)^(4/3) * n^(2/3) - n). - Vaclav Kotesovec, May 29 2019

A380639 Expansion of e.g.f. exp(x/(1 - 2*x)^2).

Original entry on oeis.org

1, 1, 9, 97, 1297, 20961, 398041, 8678209, 213337377, 5830560577, 175187949481, 5734893998241, 203021979225649, 7724154592735777, 314158263983430777, 13597375157683820161, 623802598335834369601, 30228101725367033318529, 1542430410234859308052297
Offset: 0

Views

Author

Seiichi Manyama, Jan 28 2025

Keywords

Crossrefs

Cf. A380636, A380640.
Cf. A082579.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, 2^k*binomial(2*n-k-1, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} 2^k * binomial(2*n-k-1,k)/(n-k)!.
E.g.f.: exp( Sum_{k>=1} k * 2^(k-1) * x^k ).
a(n) ~ 2^n * n^(n - 1/6) / (sqrt(3) * exp(n - 3*n^(2/3)/2 + 1/24)). - Vaclav Kotesovec, Jan 29 2025
Previous Showing 21-26 of 26 results.

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