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 11-20 of 20 results.

A361090 E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)) ).

Original entry on oeis.org

1, 1, 3, 7, -11, -239, -179, 24991, 192025, -3955391, -89483399, 552615031, 46231717621, 254468241457, -26683006147979, -571848064714289, 14926049610344881, 825004339886219521, -2973711136010539535, -1134313888244827421465, -17734152216328857754739
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A161630, A212722, A212917, A361091, A361092.
Cf. A361067, A361095.

Programs

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

Formula

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

A361091 E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)^2) ).

Original entry on oeis.org

1, 1, 3, 1, -71, -19, 10051, 12349, -3185391, -9346247, 1797304771, 9717361721, -1582301193527, -13722004186331, 2000705907453891, 25552516703201461, -3432004488804778079, -60960914621687232271, 7660860906885122096515
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A161630, A212722, A212917, A361090, A361092.
Cf. A361068, A361096.

Programs

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

Formula

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

A361092 E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)^3) ).

Original entry on oeis.org

1, 1, 3, -5, -107, 1041, 20701, -440033, -8464455, 343190593, 5639857561, -423764450889, -4968055259771, 754544622295153, 3846355902999429, -1818148417882379729, 6637679490204153841, 5658469355898945338625, -84578525845602646639823
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A161630, A212722, A212917, A361090, A361091.
Cf. A361069, A361097.

Programs

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

Formula

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

A364938 E.g.f. satisfies A(x) = exp( x / (1 - x*A(x))^3 ).

Original entry on oeis.org

1, 1, 7, 73, 1141, 23821, 623341, 19650793, 725478601, 30714824377, 1467394945561, 78103975313101, 4583805610661245, 294093243091237669, 20479664124384110101, 1538423857251845781841, 124007828871708989798161, 10676865465119963987425009
Offset: 0

Views

Author

Seiichi Manyama, Aug 14 2023

Keywords

Crossrefs

Cf. A161630, A161635.
Cf. A364940, A364942, A364981.

Programs

  • Mathematica
    Join[{1}, Table[n! * Sum[(n-k+1)^(k-1) * Binomial[n+2*k-1,n-k]/k!, {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Nov 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, (n-k+1)^(k-1)*binomial(n+2*k-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (n-k+1)^(k-1) * binomial(n+2*k-1,n-k)/k!.
a(n) ~ sqrt(s*(1 + 2*r*s) / (4 + 3*r - 12*r*s + 12*r^2*s^2 - 4*r^3*s^3)) * n^(n-1) / (exp(n) * r^n), where r = 0.1811100305436879929789759231994897963241226689... and s = 1.893740207738561813713992833266450862854198944672... are real roots of the system of equations exp(r/(1 - r*s)^3) = s, 3*s*r^2 = (1 - r*s)^4. - Vaclav Kotesovec, Nov 18 2023

A372200 E.g.f. A(x) satisfies A(x) = exp( 2 * x / (1 - x * A(x)^(1/2))^2 ).

Original entry on oeis.org

1, 2, 12, 116, 1600, 28832, 643864, 17190392, 534707296, 19003345568, 760054943464, 33798503960168, 1654577248619728, 88437537019736816, 5125378381513865752, 320163561707158120568, 21445740148760729672896, 1533498858453023915309888
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2024

Keywords

Crossrefs

Cf. A161630, A372201.
Cf. A161635, A372160.

Programs

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

Formula

E.g.f.: A(x) = B(x)^2 where B(x) is the e.g.f. of A161635.
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(n+(s-1)*k-1,n-k)/k!.

A372201 E.g.f. A(x) satisfies A(x) = exp( 3 * x / (1 - x * A(x)^(1/3))^3 ).

Original entry on oeis.org

1, 3, 27, 351, 6309, 145143, 4083669, 136159299, 5256248265, 230783968395, 11364265672929, 620524946670687, 37222254648712989, 2433741005377774719, 172301622840992025117, 13133140607475128862747, 1072406955985984437773841, 93406430850089038192704915
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2024

Keywords

Crossrefs

Cf. A161630, A372200.
Cf. A364938.

Programs

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

Formula

E.g.f.: A(x) = B(x)^3 where B(x) is the e.g.f. of A364938.
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(n+(s-1)*k-1,n-k)/k!.

A382016 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 3, 37, 901, 32141, 1502701, 86737645, 5952271977, 473117681881, 42731313784921, 4321503662185601, 483709266378568429, 59360036142346311685, 7924411424305558028757, 1143251381667547987358581, 177245340974472998607370321, 29386977237154379581209716657
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A161629, A161630, A382015.
Cf. A002293, A380513.

Programs

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

Formula

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

A372160 E.g.f. A(x) satisfies A(x) = exp( 2 * x / (1 - x * A(x)^(1/2)) ).

Original entry on oeis.org

1, 2, 8, 56, 568, 7592, 126364, 2522060, 58760272, 1566368432, 47036927284, 1571615915828, 57841636573912, 2325362549256008, 101399801919677356, 4767244262108645948, 240395075369097851296, 12943276401835227578720, 741127491503124866498404
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2024

Keywords

Crossrefs

Cf. A161630.

Programs

  • Mathematica
    terms=19; A[]=1; Do[A[x] = Exp[2*x/(1-x*Sqrt[A[x]])] + O[x]^terms // Normal, terms];CoefficientList[Series[A[x],{x,0,terms}],x]Range[0,terms-1]! (* Stefano Spezia, Aug 26 2025 *)
  • PARI
    a(n, r=2, s=1, t=0, u=1) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(n+(s-1)*k-1, n-k)/k!);

Formula

E.g.f.: A(x) = B(x)^2 where B(x) is the e.g.f. of A161630.
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(n+(s-1)*k-1,n-k)/k!.
a(n) ~ (1 + 2*LambertW(1/2))^(n + 1/2) * n^(n-1) / (sqrt(1 + LambertW(1/2)) * 2^(2*n+3) * exp(n) * LambertW(1/2)^(2*n + 7/2)). - Vaclav Kotesovec, Aug 27 2025

A382015 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 3, 31, 589, 16121, 574621, 25206595, 1312188249, 79030103185, 5404390242841, 413597889825011, 35018686148243029, 3249772250267517001, 327996955065621786309, 35769289851588288786211, 4191277822883571632163121, 525144087149768803822788257, 70060367710090279786176259633
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A161629, A161630, A382016.
Cf. A001764, A251569.

Programs

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

Formula

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

A372161 E.g.f. A(x) satisfies A(x) = exp( 3 * x / (1 - x * A(x)^(1/3)) ).

Original entry on oeis.org

1, 3, 15, 117, 1269, 17763, 305829, 6264261, 148974009, 4037901219, 122940227169, 4155745911837, 154473245377317, 6263647154467875, 275184369838089357, 13023134386197318837, 660560328648108969201, 35751895401064184128707
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2024

Keywords

Crossrefs

Cf. A161630.

Programs

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

Formula

E.g.f.: A(x) = B(x)^3 where B(x) is the e.g.f. of A161630.
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(n+(s-1)*k-1,n-k)/k!.
Previous Showing 11-20 of 20 results.

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

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