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.

A377548 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x))^3 ).

Original entry on oeis.org

1, 3, 36, 789, 25644, 1112655, 60584058, 3975599271, 305587795320, 26941234079259, 2680537845979470, 297158198268036963, 36325021999771692036, 4854553774172042934279, 704185171457954845825026, 110192472149320674192100815, 18503193203651913813111781488, 3318723221891108953801703239731
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2024

Keywords

Links

Crossrefs

Cf. A213644, A377546.
Cf. A365177.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x*exp(x))^3)/x))
  • PARI
    a(n) = 3*n!*sum(k=0, n, k^(n-k)*binomial(3*n+k+3, k)/((3*n+k+3)*(n-k)!));

Formula

E.g.f. satisfies A(x) = 1/(1 - x * A(x) * exp(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A365177.
a(n) = 3 * n! * Sum_{k=0..n} k^(n-k) * binomial(3*n+k+3,k)/( (3*n+k+3)*(n-k)! ).

A377629 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x))^4 ).

Original entry on oeis.org

1, 4, 60, 1644, 66712, 3611620, 245284344, 20071928212, 1923688610400, 211438912978692, 26225665058289640, 3624147718351890004, 552229557439437084816, 91990834731657653530180, 16632301623786709606057368, 3243982650658692575922907860, 678932992008068232965498759104
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Links

Crossrefs

Cf. A213644, A377546, A377548.
Cf. A377631, A377632.

Programs

  • PARI
    a(n) = 4*n!*sum(k=0, n, k^(n-k)*binomial(4*n+k+4, k)/((4*n+k+4)*(n-k)!));

Formula

E.g.f. satisfies A(x) = 1/(1 - x * A(x) * exp(x*A(x)))^4.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377631.
a(n) = 4 * n! * Sum_{k=0..n} k^(n-k) * binomial(4*n+k+4,k)/( (4*n+k+4)*(n-k)! ).

A382058 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 67, 1465, 44541, 1735681, 82527439, 4632741905, 299875704697, 21989097804961, 1801520077445331, 163092373817762137, 16168084561101716725, 1741946677697976052577, 202668693570279026375671, 25324088113475137179021601, 3382305512670022948599733233, 480858973986045019386825360577
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Cf. A161629, A382059.
Cf. A161635, A382032.
Cf. A001764, A377546, A382033.

Programs

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

Formula

Let F(x) be the e.g.f. of A377546. F(x) = log(A(x))/x = B(x*A(x))^2.
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(x))^2 ) ).
a(n) = 2 * n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.

A379859 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) * (1 - x*exp(x))^2 ).

Original entry on oeis.org

1, 1, 9, 110, 2121, 53834, 1720105, 66197578, 2984752113, 154358553986, 9009411908001, 585917934419498, 42018536835853369, 3294423846094650658, 280362373171289449209, 25739124908062020925034, 2535728977438902352557921, 266836955238122741966767874, 29872121613650590137264191665
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Links

Crossrefs

Cf. A377546, A379860.
Cf. A379684.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*exp(x)*(1-x*exp(x))^2)/x))
  • PARI
    a(n) = n!*sum(k=0, n, (-n+k-1)^(n-k)*binomial(2*n+k+1, k)/(n-k)!)/(n+1);

Formula

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

A379860 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) * (1 - x*exp(x))^2 ).

Original entry on oeis.org

1, 3, 33, 670, 20201, 813626, 41138953, 2507380618, 179034345393, 14663636270146, 1355499957188321, 139617725163885002, 15858083818590019993, 1969242291969058135810, 265431275379747754496409, 38595876183118645455281386, 6022354171062480540156895457, 1003753282859589405272849735810
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Links

Crossrefs

Cf. A377546, A379859.
Cf. A377890.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*exp(-x)*(1-x*exp(x))^2)/x))
  • PARI
    a(n) = n!*sum(k=0, n, (n+k+1)^(n-k)*binomial(2*n+k+1, k)/(n-k)!)/(n+1);

Formula

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

A381478 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x) * cosh(x * A(x)) )^2.

Original entry on oeis.org

1, 2, 14, 186, 3696, 98290, 3283920, 132311354, 6246905728, 338374946466, 20688891816960, 1409607482926522, 105914955915952128, 8701156803022552466, 775923181679913938944, 74646655589398509637050, 7706371729268071660093440, 849834260414107910987980354
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A185951, A377546, A381387, A381449, A381477.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = 2*sum(k=0, n, k!*binomial(2*n+k+2, k)/(2*n+k+2)*a185951(n, k));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381477.
a(n) = 2 * Sum_{k=0..n} k! * binomial(2*n+k+2,k)/(2*n+k+2) * A185951(n,k).
E.g.f.: (1/x) * Series_Reversion( x*(1 - x*cosh(x))^2 ).
Showing 1-6 of 6 results.

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