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

A349559 E.g.f. satisfies A(x) = 1/(1 - x*A(x))^x.

Original entry on oeis.org

1, 0, 2, 3, 44, 270, 3714, 44940, 746528, 13149864, 271954440, 6154715160, 155055594792, 4254730262640, 127019898548256, 4088313657038520, 141301521555548160, 5216698433745207360, 204946906542573645504, 8536144551987171202560
Offset: 0

Views

Author

Seiichi Manyama, Nov 22 2021

Keywords

Crossrefs

Cf. A007113, A349556.
Cf. A184949, A356786, A356787.

Programs

  • Maple
    a:= n-> n!*coeff(series(RootOf(1/(1-x*A)^x-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 22 2021
  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x*A[x])^x + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 22 2021 *)
  • PARI
    my(A=1,n=22); for(i=1, n, A=(1-x*A)^(-x+x*O(x^n))); Vec(serlaplace(A))
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, Aug 27 2022

Formula

a(n) ~ sqrt(1 + r - 2*r*log(r)) * n^(n-1) / ((1+r)^2 * exp(n) * r^(n + 1/2)), where r = 0.4214518303433019663622598075106479936652984008256... is the root of the equation r^(1-r) * (1+r)^(1+r) = 1. - Vaclav Kotesovec, Nov 22 2021
a(n) = n! * Sum_{k=0..floor(n/2)} (n-k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, Aug 27 2022

A355766 E.g.f. satisfies A(x) = 1/(1 - x*A(x))^(A(x)^2).

Original entry on oeis.org

1, 1, 8, 126, 3028, 98540, 4056948, 202301456, 11855415920, 798682318848, 60823290655680, 5167260183157248, 484519323081722784, 49705696509114472320, 5537956421036240838336, 665926312161296782156800, 85960998514145805006711552
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2022

Keywords

Links

Crossrefs

Cf. A001761, A349556.

Programs

  • Mathematica
    Table[Sum[(n + 2*k + 1)^(k - 1)* Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 21 2022 *)
  • PARI
    a(n) = sum(k=0, n, (n+2*k+1)^(k-1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (n+2*k+1)^(k-1) * |Stirling1(n,k)|.
a(n) ~ s^(5/2) * n^(n-1) * sqrt((1 - r*s)/(2 - 4*r*s + 2*r^2*s^2 + 3*r*s^3 - 2*r^2*s^4)) / (exp(n) * r^(n - 1/2)), where r = 0.11275303067590951818975824... and s = 1.382171434168172073998532... are real roots of the system of equations (1 - r*s)^(s^2) = 1/s, r*s/(1 - r*s) = 1/s^2 + 2*log(1 - r*s). - Vaclav Kotesovec, Jul 21 2022

A356960 E.g.f. satisfies: A(x) = 1/(1 - x * A(x)^3)^A(x).

Original entry on oeis.org

1, 1, 10, 207, 6620, 288040, 15891234, 1063219640, 83665143176, 7572321823536, 775010639465040, 88510236140283672, 11158965455394331992, 1539441941412714237912, 230675631266761375815288, 37309025609545822539225240, 6478248637390494598048444224
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2022

Keywords

Links

Crossrefs

Cf. A052813, A349556, A356914.
Cf. A356973.

Programs

  • Mathematica
    a[n_] := Sum[(3*n + k + 1)^(k - 1)*Abs[StirlingS1[n, k]], {k, 0, n}] (* Sidney Cadot, Jan 05 2023 *)
  • PARI
    a(n) = sum(k=0, n, (3*n+k+1)^(k-1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (3*n+k+1)^(k-1) * |Stirling1(n,k)|.

A356967 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x^2/2 * A(x)).

Original entry on oeis.org

1, 0, 0, 3, 6, 20, 540, 4284, 35700, 698760, 10591560, 155024100, 3231336240, 68704908480, 1451024062488, 35843008924800, 950925111973200, 25881184257364800, 765471782499860160, 24210772652940214320, 797021694060934204800, 27752674398657594805440
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Crossrefs

Cf. A184949, A349556, A356968.
Cf. A356752, A356962.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (n-k+1)^(k-1) * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).

A356968 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x^3/6 * A(x)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 4144, 40320, 409800, 4527600, 72552480, 1170449280, 19489513680, 338983444800, 6672681818880, 141166715289600, 3149324442700800, 73497460049395200, 1825098639493104000, 47984287767342796800, 1326460667797094860800
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Crossrefs

Cf. A184949, A349556, A356967.
Cf. A356753, A356963.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\4, (n-2*k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (n-2*k+1)^(k-1) * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).

A357343 E.g.f. satisfies A(x) = -log(1 - x * exp(A(x))) * exp(A(x)).

Original entry on oeis.org

0, 1, 5, 53, 878, 19904, 573984, 20112770, 829953368, 39425517072, 2119169565120, 127163052628512, 8426599011632592, 611181716437826832, 48159349246147915944, 4096752391897622411880, 374189567290578072309504, 36525100459236757201316352
Offset: 0

Views

Author

Seiichi Manyama, Sep 25 2022

Keywords

Links

Crossrefs

Cf. A006963, A357344, A357345, A357422.
Cf. A052807, A349556.

Programs

  • PARI
    a(n) = sum(k=1, n, (n+k)^(k-1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=1..n} (n+k)^(k-1) * |Stirling1(n,k)|.
E.g.f.: Series_Reversion( exp(-x) * (1 - exp(-x * exp(-x))) ). - Seiichi Manyama, Sep 09 2024

A356914 E.g.f. satisfies: A(x) = 1/(1 - x * A(x)^2)^A(x).

Original entry on oeis.org

1, 1, 8, 129, 3188, 106820, 4530174, 232744274, 14054237000, 975665799792, 76568358325440, 6703512043934112, 647772783078516600, 68484470168404178928, 7863452609057425825704, 974474039841523412062440, 129636661193226128284609344, 18427094124309886080021680640
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2022

Keywords

Crossrefs

Cf. A052813, A349556, A356960.
Cf. A355766, A356972.

Programs

  • Mathematica
    m = 18; (* number of terms *)
A[_] = 0;
Do[A[x_] = 1/(1 - x*A[x]^2)^A[x] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m-1]! (* Jean-François Alcover, Sep 12 2022 *)
  • PARI
    a(n) = sum(k=0, n, (2*n+k+1)^(k-1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (2*n+k+1)^(k-1) * |Stirling1(n,k)|.

A367200 E.g.f. satisfies A(x) = 1/(1 - x*A(x))^(A(x)^3).

Original entry on oeis.org

1, 1, 10, 201, 6216, 261300, 13923762, 899679144, 68368215528, 5975493315264, 590584798045440, 65132133511182408, 7929604511136220536, 1056369806709005923992, 152854816585283562807288, 23873492748049623945947160, 4002988479012229417182249408
Offset: 0

Views

Author

Seiichi Manyama, Nov 10 2023

Keywords

Crossrefs

Cf. A264408, A367160.
Cf. A349556, A355766.
Cf. A367199, A367201.

Programs

  • PARI
    a(n) = sum(k=0, n, (n+3*k+1)^(k-1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (n+3*k+1)^(k-1) * |Stirling1(n,k)|.

A356970 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x^2 * A(x)).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 1980, 16128, 136080, 4224960, 70943040, 1087178400, 31274100000, 784834652160, 18115033128192, 565994928945600, 18161466717139200, 560655551681971200, 20108422243585658880, 769928646324249699840, 29464638272901949824000
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2022

Keywords

Crossrefs

Cf. A184949, A349556, A356971.
Cf. A356967.

Programs

  • Mathematica
    m = 21; (* number of terms *)
A[_] = 0;
Do[A[x_] = 1/(1 - x*A[x])^(x^2*A[x]) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, (n-k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (n-k+1)^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.

A356971 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x^3 * A(x)).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, 108864, 1149120, 12160800, 138045600, 5605649280, 122049607680, 2378318604480, 45712559692800, 1529842399303680, 47673689320857600, 1382823169839820800, 38831806109898547200, 1378613101427645184000
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2022

Keywords

Crossrefs

Cf. A184949, A349556, A356970.
Cf. A356968.

Programs

  • Mathematica
    m = 21; (* number of terms *)
A[_] = 0;
Do[A[x_] = 1/(1 - x*A[x])^(x^3*A[x]) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (n-2*k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (n-2*k+1)^(k-1) * |Stirling1(n-3*k,k)|/(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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