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.

Previous Showing 11-18 of 18 results.

A379868 E.g.f. A(x) satisfies A(x) = exp(-x*A(x)^2) + x*A(x)^2.

Original entry on oeis.org

1, 0, 1, -1, 25, -101, 2281, -19895, 472305, -6760297, 177126121, -3578690435, 105341330953, -2743981145933, 91092111623241, -2888769295882111, 107832291781283809, -4009180998104138321, 167254334458983887689, -7105017992715364001147, 328862774630320838523321
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Crossrefs

Cf. A108919, A379871.
Cf. A377859, A379879.
Cf. A364982, A367866.

Programs

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

Formula

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

A379456 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + x*exp(x)) ).

Original entry on oeis.org

1, 2, 13, 151, 2573, 58221, 1648345, 56138461, 2236816825, 102135829609, 5259937376141, 301678137203433, 19072415186892325, 1317869007328182349, 98818139178323981473, 7991908824553634264101, 693473520767940388417265, 64266613784795934251538513
Offset: 0

Views

Author

Seiichi Manyama, Dec 30 2024

Keywords

Crossrefs

Cf. A088690, A379846, A379847.
Cf. A108919, A161633.
Cf. A379699, A379700.

Programs

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

Formula

a(n) = (n!/(n+1)) * Sum_{k=0..n} (2*n-k+1)^k * binomial(n+1,n-k)/k!.
E.g.f. A(x) satisfies A(x) = exp(x*A(x)) / ( 1 - x*exp(2*x*A(x)) ). - Seiichi Manyama, Feb 04 2025

A335945 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 + x)).

Original entry on oeis.org

1, 1, 1, 4, 17, 116, 907, 9010, 102097, 1348408, 19939571, 330204854, 6015657529, 120016789348, 2597201945899, 60667591974826, 1520434054966433, 40710815980598000, 1159627208850209251, 35018022339726428926, 1117395892399939407241, 37569709612314269554396
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 01 2020

Keywords

Links

Crossrefs

Cf. A052868, A060356, A108919.

Programs

  • Mathematica
    nmax = 21; A[] = 0; Do[A[x] = Exp[x A[x]/(1 + x)] + O[x]^(nmax + 1) // Normal, nmax + 1];CoefficientList[A[x], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-(1 + x) LambertW[-x/(1 + x)]/x, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] (k + 1)^(k - 1) n!/k!, {k, 0, n}], {n, 0, 21}]
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1+x))))) \\ Seiichi Manyama, Mar 05 2023

Formula

E.g.f.: -(1 + x) * LambertW(-x/(1 + x)) / x.
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * (k+1)^(k-1) * n! / k!.
a(n) ~ (exp(1) - 1)^(n + 1/2) * n^(n-1) / exp(n - 1/2). - Vaclav Kotesovec, Jul 01 2020
E.g.f.: exp ( -LambertW(-x/(1+x)) ). - Seiichi Manyama, Mar 05 2023

A377373 Expansion of e.g.f. (1/x) * Series_Reversion( x / (exp(-x) + 2*x) ).

Original entry on oeis.org

1, 1, 3, 14, 93, 794, 8335, 103774, 1496313, 24525458, 450478131, 9166307798, 204692557333, 4977320639290, 130918278855351, 3703846153114574, 112155490349101041, 3619411771703973410, 124011196515200953819, 4496024219722304736070, 171963129575721708667341
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2024

Keywords

Links

Crossrefs

Cf. A108919, A377374.
Cf. A376106.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x/(1-2*x))/x))
  • PARI
    a(n) = n!*sum(k=0, n, (-1)^k*2^(n-k)*(k+1)^(k-1)*binomial(n, k)/k!);

Formula

E.g.f.: (1/x) * LambertW(x / (1 - 2*x)).
a(n) = n! * Sum_{k=0..n} (-1)^k * 2^(n-k) * (k+1)^(k-1) * binomial(n,k)/k!.
a(n) = A376106(n+1)/(n+1).

A377374 Expansion of e.g.f. (1/x) * Series_Reversion( x / (exp(-x) + 3*x) ).

Original entry on oeis.org

1, 2, 9, 65, 653, 8439, 133609, 2506727, 54408633, 1341637595, 37055451101, 1133391705819, 38034022035877, 1389484163236727, 54899323023464529, 2332723285215012479, 106076669681270501105, 5140202768545661266227, 264427503283923495485221, 14392750805365239040586051
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2024

Keywords

Links

Crossrefs

Cf. A108919, A377373.
Cf. A376107.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x/(1-3*x))/x))
  • PARI
    a(n) = n!*sum(k=0, n, (-1)^k*3^(n-k)*(k+1)^(k-1)*binomial(n, k)/k!);

Formula

E.g.f.: (1/x) * LambertW(x / (1 - 3*x)).
a(n) = n! * Sum_{k=0..n} (-1)^k * 3^(n-k) * (k+1)^(k-1) * binomial(n,k)/k!.
a(n) = A376107(n+1)/(n+1).

A379701 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) / (1 + x*exp(2*x)) ).

Original entry on oeis.org

1, 0, 3, 2, 113, 304, 13747, 83600, 3590337, 38193920, 1650383171, 26535997696, 1186785903217, 26244849422336, 1234578346302771, 35176362803984384, 1757110507998276353, 61533880908307038208, 3281634015502670522371, 136392534106346468999168
Offset: 0

Views

Author

Seiichi Manyama, Dec 30 2024

Keywords

Crossrefs

Cf. A108919, A379702.
Cf. A366232.

Programs

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

Formula

a(n) = (n!/(n+1)) * Sum_{k=0..n} (n-2*k-1)^k * binomial(n+1,n-k)/k!.

A379702 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) / (1 + x*exp(3*x)) ).

Original entry on oeis.org

1, 0, 5, 11, 333, 2829, 78553, 1360197, 42149817, 1123709129, 40775629581, 1453036152897, 62005204699045, 2736440768515869, 135913168259011809, 7106229274104610829, 405068417020871464689, 24398077807975709138193, 1574189366334360310720405
Offset: 0

Views

Author

Seiichi Manyama, Dec 30 2024

Keywords

Crossrefs

Cf. A108919, A379701.
Cf. A366233.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*n-3*k-1)^k*binomial(n+1, n-k)/k!)/(n+1);

Formula

a(n) = (n!/(n+1)) * Sum_{k=0..n} (2*n-3*k-1)^k * binomial(n+1,n-k)/k!.

A379866 Expansion of e.g.f. (1/x) * Series_Reversion( x / (exp(-x) + x)^2 ).

Original entry on oeis.org

1, 0, 2, -2, 56, -222, 5332, -45782, 1127408, -15972542, 428055644, -8598013734, 256717806952, -6667767637598, 223389539254676, -7076616268104278, 265762684840216544, -9880557234248622462, 413902270494309471436, -17591536945041528005318, 816621849842712202724696
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Links

Crossrefs

Cf. A108919, A367868.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = (exp(-x*A(x)) + x*A(x))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A379868.
a(n) = -2 * n! * Sum_{k=0..n} (-2*n+k-2)^(n-k-1) * binomial(2*n+1,k)/(n-k)!.
Previous Showing 11-18 of 18 results.

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