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 1-9 of 9 results.

A377890 E.g.f. A(x) satisfies A(x) = (1 + x*A(x)^2) * exp(x * A(x)).

Original entry on oeis.org

1, 2, 15, 211, 4433, 124741, 4412815, 188335981, 9421966209, 540884623753, 35054089163351, 2531882857204273, 201689970517618225, 17567711167993834381, 1661084543502646535967, 169448367505003640681221, 18550123929621138841581185, 2169272360350263071212545553
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2024

Keywords

Crossrefs

Cf. A088690, A377826, A377891.

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!.
E.g.f.: (1/x) * Series_Reversion( x * (exp(-x) - x) ). - Seiichi Manyama, Dec 29 2024

A377827 E.g.f. satisfies A(x) = (1 + x)^2 * exp(x * A(x)).

Original entry on oeis.org

1, 3, 13, 106, 1273, 20226, 402589, 9637902, 269967793, 8666441650, 313793596981, 12653878751526, 562489374836041, 27328756018660266, 1440892788988703821, 81940739770677315646, 4999648556871348611425, 325806859913842861709922, 22584652022005415601772645
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A377826, A377828.
Cf. A362772, A377740, A377810.

Programs

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

Formula

E.g.f.: (1+x)^2 * exp( -LambertW(-x*(1+x)^2) ).
E.g.f.: -LambertW(-x*(1+x)^2)/x.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(2*k+2,n-k)/k!.
a(n) ~ sqrt(1 + 3*r) * n^(n-1) / (exp(n - 1/4) * r^(n + 3/4)), where r = 0.2394629861788505554394435808448... is root of the equation r*(1+r)^2 = exp(-1). - Vaclav Kotesovec, Nov 09 2024

A377828 E.g.f. satisfies A(x) = (1 + x)^3 * exp(x * A(x)).

Original entry on oeis.org

1, 4, 21, 193, 2669, 48711, 1113325, 30615019, 984983193, 36319515355, 1510538562641, 69968975169567, 3572684914283941, 199389519518767111, 12075888110164192917, 788850329621989132771, 55289606764547108653361, 4138807268239824817387443, 329564746571982961088975257
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A377826, A377827.
Cf. A377741, A377811.

Programs

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

Formula

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

A377894 E.g.f. satisfies A(x) = (1 + x) * exp(x * A(x)^2).

Original entry on oeis.org

1, 2, 11, 142, 2725, 71026, 2339719, 93311758, 4371948137, 235418287042, 14327098759171, 972533690209390, 72854996624174989, 5970582808814848498, 531359818098465084863, 51034785131352404960686, 5261620527219949295345233, 579593410301187097865649922
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2024

Keywords

Links

Crossrefs

Cf. A377826, A377895.

Programs

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

Formula

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

A377895 E.g.f. satisfies A(x) = (1 + x) * exp(x * A(x)^3).

Original entry on oeis.org

1, 2, 15, 283, 8057, 313161, 15436735, 922964771, 64910124753, 5250807814753, 480339263735831, 49032749858906067, 5525542086267361801, 681359718334607629409, 91259859216031641999375, 13193464971338727171704611, 2047721360761921797402720545, 339610337568547449759788735553
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2024

Keywords

Links

Crossrefs

Cf. A377826, A377894.

Programs

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

Formula

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

A378045 E.g.f. satisfies A(x) = (1+x) * exp(x * A(x)^2 / (1+x)).

Original entry on oeis.org

1, 2, 9, 100, 1693, 39046, 1140589, 40379872, 1680490361, 80409242314, 4349556199441, 262478904794140, 17482853419143061, 1274026039224276430, 100830973069183104245, 8612770277501109271576, 789749958006001265241073, 77375794118912255978104978, 8066966112797470401673208089
Offset: 0

Views

Author

Seiichi Manyama, Nov 15 2024

Keywords

Links

Crossrefs

Cf. A377826, A378046.
Cf. A377894, A378043.
Cf. A362773.

Programs

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

Formula

E.g.f.: (1+x) * exp( -LambertW(-2*x*(1+x))/2 ).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(k+1,n-k)/k!.
a(n) ~ sqrt(1 + 2*exp(-1) - sqrt(1 + 2*exp(-1))) * (1 + sqrt(1 + 2*exp(-1))) * 2^(n-2) * n^(n-1) / ((sqrt(1 + 2*exp(-1)) - 1)^n * exp(n-1)). - Vaclav Kotesovec, Nov 15 2024

A378046 E.g.f. satisfies A(x) = (1+x) * exp(x * A(x)^3 / (1+x)^2).

Original entry on oeis.org

1, 2, 11, 169, 4049, 132881, 5542495, 280694135, 16730578625, 1147444968385, 89015365063991, 7707022678811567, 736734708409976017, 77070404075178587633, 8757816984586841345231, 1074244834335107678837191, 141469329806979182825146625, 19908315372027482035799282177
Offset: 0

Views

Author

Seiichi Manyama, Nov 15 2024

Keywords

Links

Crossrefs

Cf. A377826, A378045.
Cf. A363478.

Programs

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

Formula

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

A378047 E.g.f. satisfies A(x) = (1+x)^2 * exp(x * A(x) / (1+x)).

Original entry on oeis.org

1, 3, 11, 70, 685, 8966, 147769, 2938608, 68509193, 1832813866, 55354862701, 1863179311676, 69163696768093, 2807246931378462, 123685264726805825, 5879149578658117096, 299892675674572370065, 16340561709320173229906, 947234845622653951286485
Offset: 0

Views

Author

Seiichi Manyama, Nov 15 2024

Keywords

Links

Crossrefs

Cf. A362771, A377826.

Programs

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

Formula

E.g.f.: (1+x)^2 * exp( -LambertW(-x*(1+x)) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(k+2,n-k)/k!.
a(n) ~ (1 + sqrt(1 + 4*exp(-1)))^2 * sqrt(2 + 8*exp(-1) - 2*sqrt(1 + 4*exp(-1))) * 2^(n-3) * n^(n-1) / ((sqrt(1 + 4*exp(-1)) - 1)^n * exp(n - 3/2)). - Vaclav Kotesovec, Nov 15 2024
Showing 1-9 of 9 results.

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

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