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

A379661 Expansion of e.g.f. (1/x) * Series_Reversion( x * (exp(-2*x) - x) ).

Original entry on oeis.org

1, 3, 32, 638, 18992, 757152, 37946944, 2294428880, 162614380544, 13224995258624, 1214230905887744, 124244350629669888, 14021254268885487616, 1730161184870597414912, 231756218205467830255616, 33492293939832097344370688, 5194237715163522538324557824, 860510764365813120075516739584
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2024

Keywords

Crossrefs

Cf. A377890, A379662.

Programs

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

Formula

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

A379867 E.g.f. A(x) satisfies A(x) = 1/(exp(-x*A(x)^2) - x*A(x)^2).

Original entry on oeis.org

1, 2, 23, 529, 18589, 884281, 53195407, 3874595089, 331580316473, 32614443047521, 3625839880813171, 449629404853604185, 61535275741655857621, 9213155228282408405185, 1498018121369750569371959, 262869047482982449625840161, 49515850496472530668242845041
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Crossrefs

Cf. A377890, A379870.
Cf. A377892, A379884.
Cf. A377891, A377893, A379886.
Cf. A364985, A379864.

Programs

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

Formula

E.g.f.: sqrt( (1/x) * Series_Reversion( x * (exp(-x) - x)^2 ) ).
a(n) = n! * Sum_{k=0..n} (3*n-k+1)^(k-1) * binomial(3*n-k+1,n-k)/k!.

A377891 E.g.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * exp(x * A(x)).

Original entry on oeis.org

1, 2, 19, 364, 10665, 423056, 21221851, 1288931456, 91977076561, 7543664425216, 699290913249891, 72306463481715200, 8251192866018497401, 1030074741274860240896, 139650729116792108398891, 20432888021354725476499456, 3209204194084043665909835937, 538542735919965101952197525504
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2024

Keywords

Crossrefs

Cf. A088690, A377826, A377890.

Programs

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

Formula

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

A379659 Expansion of e.g.f. (1/x) * Series_Reversion( x * (exp(-x) - 2*x) ).

Original entry on oeis.org

1, 3, 35, 766, 25037, 1096026, 60318343, 4004850718, 311682426329, 27835056557650, 2806352022690971, 315328008662299062, 39076741980040016293, 5294976627701723411866, 778852468463039787264911, 123599090729382829502733646, 21049364262909179898132228017, 3829309039543060842371185060770
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2024

Keywords

Crossrefs

Cf. A000272, A377890, A379660.

Programs

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

Formula

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

A379870 E.g.f. A(x) satisfies A(x) = 1/(exp(-x*A(x)^3) - x*A(x)^3).

Original entry on oeis.org

1, 2, 31, 991, 48873, 3276921, 278486359, 28694553119, 3476833863281, 484490228040865, 76339085661865791, 13421203354104200271, 2604724304171427849145, 553128917492225243766065, 127578750880241791377948359, 31761039697155404251033218751, 8488576933611794321694363786849
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A377890, A379867.

Programs

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

Formula

E.g.f.: ( (1/x) * Series_Reversion( x * (exp(-x) - x)^3 ) )^(1/3).
a(n) = n! * Sum_{k=0..n} (4*n-k+1)^(k-1) * binomial(4*n-k+1,n-k)/k!.

A379684 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) * (1 - x*exp(x)) ).

Original entry on oeis.org

1, 0, 3, 11, 193, 2389, 50191, 1088205, 29836353, 902845673, 31428924631, 1207426391137, 51394833121105, 2386048646491197, 120379283952129567, 6547887322803355589, 382306453347573490177, 23839225109022069540817, 1581540933047988924532135
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2024

Keywords

Crossrefs

Cf. A213644, A377890, A379685.

Programs

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

Formula

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

A379885 E.g.f. A(x) satisfies A(x) = 1/(exp(-x) - x*A(x)).

Original entry on oeis.org

1, 2, 11, 118, 1885, 40266, 1080679, 34979134, 1326825497, 57744176914, 2836795756771, 155305155441030, 9376803979425205, 619006372481008474, 44357422104298022399, 3429215554499681260366, 284496868838293052890033, 25212167721275946619910178, 2377021703587467346833760315
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A377890, A377892.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(2/(exp(-x)+sqrt(exp(-2*x)-4*x))))
  • PARI
    a(n) = n!*sum(k=0, n, (2*n-2*k+1)^(k-1)*binomial(2*n-2*k+1, n-k)/k!);

Formula

E.g.f.: 2/(exp(-x) + sqrt(exp(-2*x) - 4*x)).
a(n) = n! * Sum_{k=0..n} (2*n-2*k+1)^(k-1) * binomial(2*n-2*k+1,n-k)/k!.
a(n) ~ sqrt(1 + LambertW(1/2)) * 2^n * n^(n-1) / (LambertW(1/2)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jan 05 2025

A379912 E.g.f. A(x) satisfies A(x) = 1/(exp(-x*A(x)^3) - x*A(x)).

Original entry on oeis.org

1, 2, 23, 541, 19585, 962901, 59969227, 4526706661, 401724516641, 40994441922169, 4729721311570411, 608827327842480825, 86507217246635276065, 13448830748996370988885, 2270847762050485928361227, 413849998079530364443224781, 80967576778854924208520130241
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Links

Crossrefs

Cf. A377893, A379870, A379897.
Cf. A377890, A377892, A379885.

Programs

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

Formula

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

A379660 Expansion of e.g.f. (1/x) * Series_Reversion( x * (exp(-x) - 3*x) ).

Original entry on oeis.org

1, 4, 63, 1861, 82097, 4850511, 360275791, 32284172215, 3391036374849, 408722465393947, 55615320933323831, 8433939560030789091, 1410592999486776429841, 257966146919938099737511, 51211682886276933924579999, 10968423456294584033965364191, 2521058533839507726700577317889
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2024

Keywords

Crossrefs

Cf. A000272, A377890, A379659.

Programs

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

Formula

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

A379662 Expansion of e.g.f. (1/x) * Series_Reversion( x * (exp(-3*x) - x) ).

Original entry on oeis.org

1, 4, 55, 1407, 53697, 2743893, 176247927, 13657260501, 1240444335969, 129281173193385, 15210977483374479, 1994549350608453609, 288445997202717914433, 45611213861740547292093, 7829287622920723803498279, 1449907335654700964735333997, 288151338120152032299063196737
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2024

Keywords

Crossrefs

Cf. A377890, A379661.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} 3^k * (2*n-k+1)^(k-1) * binomial(2*n-k+1,n-k)/k!.
Showing 1-10 of 12 results. Next

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