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 1-6 of 6 results.

A380515 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 7, 109, 2689, 91261, 3950191, 208064137, 12917499169, 923765042809, 74780847503191, 6760168138392901, 675023676995501857, 73787463232202560309, 8763902701210982610559, 1123850728979698205132641, 154757223522414820829369281, 22775744033825102490806751217
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2025

Keywords

Crossrefs

Cf. A380513, A380514, A380516.
Cf. A080893, A380511.
Cf. A091695, A250917, A380512.
Cf. A002293, A006632, A370057, A382059.

Programs

  • PARI
    a(n) = if(n==0, 1, 3*n!*sum(k=0, n-1, binomial(3*n+k, k)/((3*n+k)*(n-k-1)!)));

Formula

a(n) = 3 * n! * Sum_{k=0..n-1} binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-4*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x*(1-x)^3 ) ). - Seiichi Manyama, Mar 15 2025

A380516 Expansion of e.g.f. exp(x*G(x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 9, 157, 4129, 146001, 6502681, 349790029, 22069858497, 1598577634369, 130757736096361, 11922399644742621, 1199121973234651489, 131887738425602277457, 15748194681225620534649, 2028885239529647188594381, 280525944581514367875035521, 41434950383158772951280658689
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2025

Keywords

Links

Crossrefs

Cf. A002293, A380513, A380514, A380515.
Cf. A000262, A251568, A380512.
Cf. A382034.

Programs

  • Mathematica
    Join[{1}, Table[(n-1)! * LaguerreL[n-1, 3*n+1, -1], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 26 2025 *)
  • PARI
    a(n) = if(n==0, 1, (n-1)!*pollaguerre(n-1, 3*n+1, -1));

Formula

E.g.f.: exp(G(x)-1), where G(x) is described above.
a(n) = (n-1)! * Sum_{k=0..n-1} binomial(4*n,k)/(n-k-1)! for n > 0.
a(n+1) = n! * LaguerreL(n, 3*n+4, -1).
a(n) = (-1)^(n+1)*U(1-n, 2+3*n, -1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
a(n) ~ 2^(8*n + 1) * n^(n-1) / (exp(n - 1/3) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jan 26 2025
E.g.f.: exp( Series_Reversion( x/(1+x)^4 ) ). - Seiichi Manyama, Mar 15 2025

A380511 Expansion of e.g.f. exp(x*G(x)^2) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 55, 961, 23141, 711421, 26631235, 1175535425, 59786520841, 3442729157461, 221413508687471, 15730688410899265, 1223574846548300845, 103417508018836074701, 9437941200860641295611, 924934291227615821904001, 96881241931552168636182545, 10801002623361396194857667365
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2025

Keywords

Crossrefs

Cf. A251569, A380512.
Cf. A080893, A380515.
Cf. A082579, A251568, A380514.
Cf. A001764, A006013, A370054, A382058.

Programs

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

Formula

a(n) = 2 * n! * Sum_{k=0..n-1} binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-3*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x*(1-x)^2 ) ). - Seiichi Manyama, Mar 15 2025

A250917 Expansion of e.g.f. exp( x*C(x)^3 ) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, A000108.

Original entry on oeis.org

1, 1, 7, 73, 1033, 18541, 403831, 10351237, 305355793, 10192132153, 379819484551, 15634219476481, 704566985120857, 34506514429777573, 1825081888365736183, 103685565729559782781, 6297505655719537293601, 407233553972252986277617, 27935786938445348562454663
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2014

Keywords

Comments

In general, if k>0 and e.g.f. = exp(x*C(x)^k) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, then a(n) ~ k * 2^(2*n + k - 5/2) * n^(n-1) / exp(n - 2^(k-2)). - Vaclav Kotesovec, Aug 22 2017

Examples

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 73*x^3/3! + 1033*x^4/4! + 18541*x^5/5! +...
such that log(A(x)) = x*C(x)^3,
log(A(x)) = x + 3*x^2 + 9*x^3 + 28*x^4 + 90*x^5 + 297*x^6 + 1001*x^7 +...
where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Crossrefs

Cf. A080893, A251568.
Cf. A091695, A380512, A380515.
Cf. A000108, A001517.

Programs

  • PARI
    {a(n)=my(C=1); for(i=1, n, C=1+x*C^2 +x*O(x^n));
n!*polcoef(exp(x*C^3), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = if(n==0, 1, sum(k=0, n, n!/k! * binomial(2*n+k-1, n-k) * 3*k/(n+2*k) ))}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(serreverse(x*(1-x))^3/x^2))) \\ Seiichi Manyama, Mar 15 2025

Formula

a(n) = Sum_{k=0..n} n!/k! * binomial(2*n+k-1, n-k) * 3*k/(n+2*k) for n>0 with a(0)=1.
a(n) ~ 3 * 2^(2*n+1/2) * n^(n-1) / exp(n-2). - Vaclav Kotesovec, Aug 22 2017
Conjecture D-finite with recurrence: +2*a(n) +(-11*n+20)*a(n-1) +(n^3+9*n^2-116*n+164)*a(n-2) +(-4*n^4+35*n^3+n^2-317*n+342)*a(n-3) -6*(n-3)*(6*n^3-50*n^2+147*n-176)*a(n-4) +12*(n-5)*(2*n-9)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 25 2020
E.g.f.: exp( (1/x)^2 * Series_Reversion( x*(1-x) )^3 ). - Seiichi Manyama, Mar 15 2025

A380637 Expansion of e.g.f. exp(x*G(3*x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 19, 703, 39313, 2959921, 280935811, 32221238239, 4336213980673, 670088514363553, 116959281939738451, 22759439305951039231, 4885844614853182749649, 1147088485458553806981073, 292394958982688921734424323, 80420728320326634679448511391
Offset: 0

Views

Author

Seiichi Manyama, Jan 28 2025

Keywords

Links

Crossrefs

Cf. A001764, A380512.
Cf. A380641.

Programs

  • PARI
    a(n) = if(n==0, 1, 3^(n-1)*(n-1)!*pollaguerre(n-1, 2*n+1, -1/3));

Formula

E.g.f.: exp( (G(3*x)-1)/3 ), where G(x) is described above.
a(n) = (n-1)! * Sum_{k=0..n-1} 3^k * binomial(3*n,k)/(n-k-1)! for n > 0.
a(n+1) = 3^n * n! * LaguerreL(n, 2*n+3, -1/3).
a(n) ~ 3^(4*n - 1/2) * n^(n-1) / (2^(2*n + 3/2) * exp(n - 1/6)). - Vaclav Kotesovec, Jan 29 2025
a(n) = (-3)^(n-1)*U(1-n, 2*(1+n), -1/3), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 29 2025
E.g.f.: exp( Series_Reversion( x/(1+3*x)^3 ) ). - Seiichi Manyama, Mar 16 2025

A382101 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(B_k(x) - 1), where B_k(x) = 1 + x*B_k(x)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 43, 73, 1, 1, 1, 9, 91, 529, 501, 1, 1, 1, 11, 157, 1753, 8501, 4051, 1, 1, 1, 13, 241, 4129, 45001, 169021, 37633, 1, 1, 1, 15, 343, 8041, 146001, 1447471, 4010455, 394353, 1, 1, 1, 17, 463, 13873, 362501, 6502681, 56041987, 110676833, 4596553, 1
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Examples

			Square array begins:
  1,   1,    1,     1,      1,      1, ...
  1,   1,    1,     1,      1,      1, ...
  1,   3,    5,     7,      9,     11, ...
  1,  13,   43,    91,    157,    241, ...
  1,  73,  529,  1753,   4129,   8041, ...
  1, 501, 8501, 45001, 146001, 362501, ...

Links

Crossrefs

Columns k=0..4 give A000012, A000262, A251568, A380512, A380516.
Cf. A355262, A382100.

Programs

  • PARI
    a(n, k) = if(n==0, 1, (n-1)!*pollaguerre(n-1, (k-1)*n+1, -1));

Formula

A(n,k) = (n-1)! * Sum_{j=0..n-1} binomial(k*n,j)/(n-j-1)! for n > 0.
A(n,k) = (n-1)! * LaguerreL(n-1, (k-1)*n+1, -1) for n > 0.
E.g.f. of column k: exp( Series_Reversion( x/(1+x)^k ) ).
Showing 1-6 of 6 results.

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