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 1-10 of 12 results. Next

A364978 E.g.f. satisfies A(x) = 1 + x*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 2, 15, 124, 1565, 23886, 446887, 9787352, 246408633, 7010910010, 222438284651, 7788393551412, 298293192119221, 12406118302851014, 556817903190669135, 26825727269937929776, 1380790608848655193457, 75625529930102546486514
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A161631, A364979.

Programs

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

Formula

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

A364979 E.g.f. satisfies A(x) = 1 + x*exp(x*A(x)^3).

Original entry on oeis.org

1, 1, 2, 21, 220, 3545, 70566, 1702267, 48438104, 1582227873, 58475787850, 2410935939731, 109728296017572, 5464423604085745, 295562179335075758, 17255009243888243115, 1081438061864539992496, 72422934220506772042817, 5161269584065131270532242
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A161631, A364978.

Programs

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

Formula

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

A371044 E.g.f. satisfies A(x) = 1 + x^3*exp(x*A(x)).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 120, 5250, 80976, 726264, 4839120, 86487390, 2283242280, 42585905076, 590667519624, 10115535833130, 286758920451360, 8128299117822960, 186279550983756576, 4123388294626654134, 118916807955913504440, 4102548791571529697580
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2024

Keywords

Crossrefs

Cf. A370985, A371019, A371045, A371046.
Cf. A161631, A371042.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1 - LambertW[-E^x*x^4]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 10 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, k^(n-3*k)*binomial(n-3*k+1, k)/((n-3*k+1)*(n-3*k)!));

Formula

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

A371042 E.g.f. satisfies A(x) = 1 + x^2*exp(x*A(x)).

Original entry on oeis.org

1, 0, 2, 6, 12, 140, 1470, 10122, 114296, 1874952, 25462170, 379431470, 7546461252, 151797222876, 3066316693622, 72101615826450, 1843378516587120, 47860832586054032, 1338908395558366386, 40675047500003794902, 1282380661224172506620
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2024

Keywords

Crossrefs

Cf. A370984, A371018, A371043.
Cf. A161631, A371044.
Cf. A364978.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1 - LambertW[-E^x*x^3]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 10 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, k^(n-2*k)*binomial(n-2*k+1, k)/((n-2*k+1)*(n-2*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} k^(n-2*k) * binomial(n-2*k+1,k)/( (n-2*k+1)*(n-2*k)! ).
From Vaclav Kotesovec, Mar 10 2024: (Start)
E.g.f.: 1 - LambertW(-exp(x)*x^3)/x.
a(n) ~ sqrt(1 + LambertW(exp(-1/3)/3)) * n^(n-1) / (exp(n) * 3^(n + 1/2) * LambertW(exp(-1/3)/3)^(n+1)). (End)

A161632 E.g.f. satisfies A(x) = (1 + x*exp(x*A(x)))^2.

Original entry on oeis.org

1, 2, 6, 42, 392, 4970, 78492, 1489838, 33105648, 842437170, 24181696820, 772887702422, 27228973364232, 1048392980781770, 43802436902618604, 1973819502540516990, 95426799849067842272, 4927195390491532227170
Offset: 0

Views

Author

Paul D. Hanna, Jun 18 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 42*x^3/3! + 392*x^4/4! + 4970*x^5/5! +...
A(x)^(1/2) = 1 + x + 2*x^2/2! + 15*x^3/3! + 124*x^4/4! + 1565*x^5/5! +...

Crossrefs

Cf. A161631, A364978.

Programs

  • Mathematica
    Flatten[{1,Table[n!*Sum[Binomial[2*(n-k+1),k]/(n-k+1) * k^(n-k)/(n-k)!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    {a(n,m=1)=n!*sum(k=0,n,m*binomial(2*(n-k+m),k)/(n-k+m)*k^(n-k)/(n-k)!)}

Formula

a(n) = n!*Sum_{k=0..n} C(2*(n-k+1),k)/(n-k+1) * k^(n-k)/(n-k)!.
If A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
a(n,m) = n!*Sum_{k=0..n} m*C(2*(n-k+m),k)/(n-k+m) * k^(n-k)/(n-k)!.
E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = (1 + x*exp(x)/B(x))^2.
a(n) ~ sqrt(2*s^(3/2)*(2-5*sqrt(s)+3*s)/(2*sqrt(s)-1)) * (2*s-2*sqrt(s))^n * n^(n-1) / exp(n), where s = 3.533778497303240223520495... is the root of the equation (2-2/sqrt(s)) * log(2*(sqrt(s)-2*s+s^(3/2))) = 1. - Vaclav Kotesovec, Jan 10 2014
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364978. - Seiichi Manyama, Nov 02 2024

A377581 E.g.f. satisfies A(x) = 1 + x * exp(x*A(x)^4).

Original entry on oeis.org

1, 1, 2, 27, 340, 6485, 156486, 4532647, 155359016, 6116223465, 272369488330, 13537882005131, 742838308204092, 44605728508797469, 2909444391161677838, 204844046364505460655, 15484082153045052133456, 1250714994867101307618257, 107511883999692161772696210
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A161631, A364978, A364979.
Cf. A377579, A377580.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*n-4*k+1, k)/((4*n-4*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*n-4*k+1,k)/( (4*n-4*k+1)*(n-k)! ).

A380093 E.g.f. A(x) satisfies A(x) = sqrt( 1 + 2*x*exp(x*A(x)) ).

Original entry on oeis.org

1, 1, 1, 6, 13, 180, 501, 13720, 34777, 2014992, 2512585, 491642976, -564313947, 181714012480, -836832558275, 95473740036480, -856984734161999, 68029327826567424, -954950936641491951, 63368301861354866176, -1238053892876418633155, 74904417332353810338816
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A161631, A380094.
Cf. A380035, A380050.

Programs

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

Formula

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

A380094 E.g.f. A(x) satisfies A(x) = ( 1 + 3*x*exp(x*A(x)) )^(1/3).

Original entry on oeis.org

1, 1, 0, 7, -28, 405, -4514, 75313, -1336824, 28494793, -672782950, 17874984501, -521966931716, 16702822898749, -579928752836874, 21736834275178345, -874384126286848624, 37581186999500130321, -1718628399364227445070, 83327485224351815544925
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A161631, A380093.
Cf. A380051.

Programs

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

Formula

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

A371063 E.g.f. satisfies A(x) = 1 + x^2/2*exp(x*A(x)).

Original entry on oeis.org

1, 0, 1, 3, 6, 40, 375, 2541, 21028, 264636, 3303765, 41219695, 625493946, 10676900598, 185753808331, 3495429297465, 72963017028840, 1606964677740376, 37107535997019753, 918150959889615771, 24110308315512081550, 662150320109499176130, 19105058680403510485671
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2024

Keywords

Crossrefs

Cf. A161631, A371066.
Cf. A371042.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1 - LambertW[-E^x*x^3/2]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 10 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, k^(n-2*k)*binomial(n-2*k+1, k)/(2^k*(n-2*k+1)*(n-2*k)!));

Formula

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

A371066 E.g.f. satisfies A(x) = 1 + x^3/6*exp(x*A(x)).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 20, 175, 2296, 20244, 134520, 1016565, 13527580, 209970046, 2785823404, 33569936855, 467250784560, 8358652382760, 159820481883696, 2888819281378089, 51781860691882740, 1031576680142770930, 23237341150372569220, 543570375735294712651
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2024

Keywords

Crossrefs

Cf. A161631, A371063.
Cf. A371044.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1 - ProductLog[-E^x*x^4/6]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 10 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, k^(n-3*k)*binomial(n-3*k+1, k)/(6^k*(n-3*k+1)*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-3*k) * binomial(n-3*k+1,k)/( 6^k*(n-3*k+1)*(n-3*k)! ).
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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