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 21-30 of 34 results. Next

A380042 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2*x*exp(x*A(x)^2) ).

Original entry on oeis.org

1, 1, 5, 48, 697, 13640, 336771, 10053778, 352334753, 14183529480, 645073504435, 32715111226886, 1830671281889649, 112049330303532388, 7446824171300128811, 534068807341887943770, 41111698162393482004801, 3381089519620006418116976, 295869084136630532211207843
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2025

Keywords

Crossrefs

Cf. A380015, A380035.
Cf. A161633, A380043.
Cf. A201470.

Programs

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

Formula

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

A380826 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x) / (1 + x*exp(-2*x)) ).

Original entry on oeis.org

1, 4, 43, 810, 22273, 811728, 36979467, 2025462736, 129748802401, 9522843081984, 788169731306059, 72641846664240384, 7379343546762675873, 819269203286474309632, 98698960328223628470379, 12824232015954542746048512, 1787731339345567827140060737, 266157254062414638948185210880
Offset: 0

Views

Author

Seiichi Manyama, Feb 04 2025

Keywords

Links

Crossrefs

Cf. A088690, A161633, A380808.
Cf. A361182, A380829, A380830.

Programs

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

Formula

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

A201470 E.g.f. satisfies: A(x) = 1/(1 - 2*x*exp(x*A(x))).

Original entry on oeis.org

1, 2, 12, 126, 1928, 39050, 987852, 30028670, 1067161104, 43439950098, 1993658601620, 101873148358982, 5736946141694616, 353052289411248986, 23574446170669354716, 1697657229173802582030, 131156091046113794979872, 10821153944570302041170978, 949646768024669592457251108
Offset: 0

Views

Author

Paul D. Hanna, Dec 01 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 126*x^3/3! + 1928*x^4/4! + 39050*x^5/5! +...
The exponential of the e.g.f. begins:
exp(x*A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 721*x^4/4! + 14241*x^5/5! +...
The coefficients of x^n/n! in the powers of G(x) = 1 + 2*x*exp(x) begin:
G^1: [(1), 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...];
G^2: [1,(4), 16, 60, 208, 660, 1944, 5404, 14368, 36900 ...];
G^3: [1, 6,(36), 210, 1176, 6270, 31716, 152250, 696240, ...];
G^4: [1, 8, 64, (504), 3872, 28840, 207408, 1436792, ...];
G^5: [1, 10, 100, 990, (9640), 91890, 854460, 7731430, ...];
G^6: [1, 12, 144, 1716, 20208,(234300), 2666952, 29736084, ...];
G^7: [1, 14, 196, 2730, 37688, 514150, (6914964), 91510034, ...];
G^8: [1, 16, 256, 4080, 64576, 1012560, 15698016,(240229360), ...]; ...
where the coefficients in parenthesis form initial terms of this sequence:
[1/1, 4/2, 36/3, 504/4, 9640/5, 234300/6, 6914964/7, 240229360/8, ...].

Crossrefs

Cf. A161633.

Programs

  • Mathematica
    CoefficientList[1/x*InverseSeries[Series[x/(1 + 2*x*Exp[x]), {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 12 2014 *)
  • PARI
    a(n, m=1)=n!*sum(k=0, n, 2^k*binomial(n+m, k)*m/(n+m)*k^(n-k)/(n-k)!)

Formula

E.g.f.: A(x) = 1 + 2*x*A(x)*exp(x*A(x)).
E.g.f.: A(x) = (1/x)*Series_Reversion[x/(1 + 2*x*exp(x))].
a(n) = [x^n/n!] (1 + 2*x*exp(x))^(n+1)/(n+1).
a(n) = n!*Sum_{k=0..n} 2^k * C(n+1,k)/(n+1) * k^(n-k)/(n-k)!.
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
a(n,m) = n!*Sum_{k=0..n} 2^k * C(n+m,k)*m/(n+m) * k^(n-k)/(n-k)!.
a(n) ~ s/sqrt(2*s-1) * n^(n-1) * ((s-1)*s)^(n+1/2) / exp(n), where s = 2.8524169182445218... is the root of the equation (s-1)*LambertW((s-1)/2) = 1. - Vaclav Kotesovec, Jan 12 2014

A370889 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x^2/2)) ).

Original entry on oeis.org

1, 1, 2, 9, 72, 735, 9000, 133035, 2325120, 46631025, 1053108000, 26484495345, 734652737280, 22280390827695, 733335188826240, 26035824337798275, 991872319953715200, 40360728513989909025, 1747119524427614937600, 80166580022376802179225
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2024

Keywords

Links

Crossrefs

Cf. A161633, A370926.
Cf. A365283.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x*exp(x^2/2)))/x))
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n+1, n-2*k)/(2^k*k!))/(n+1);

Formula

a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n+1,n-2*k)/(2^k * k!).
a(n) ~ (1 + 3*LambertW(1/3))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(1/3)) * 3^(3*n/2 + 2) * exp(n) * LambertW(1/3)^(3*(n+1)/2)). - Vaclav Kotesovec, Mar 06 2024

A370926 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x^3/6)) ).

Original entry on oeis.org

1, 1, 2, 6, 28, 220, 2520, 34510, 519680, 8527680, 154831600, 3151456000, 71830281600, 1809141934600, 49559087177600, 1459865188782000, 45970426027926400, 1543274016213529600, 55120521154277779200, 2088917638216953544000, 83717918489664018560000
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2024

Keywords

Links

Crossrefs

Cf. A161633, A370889.
Cf. A365287.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x*exp(x^3/6)))/x))
  • PARI
    a(n) = n!*sum(k=0, n\3, (n-3*k)^k*binomial(n+1, n-3*k)/(6^k*k!))/(n+1);

Formula

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

A377541 E.g.f. satisfies A(x) = 1/(1 - x * exp(x*A(x)))^2.

Original entry on oeis.org

1, 2, 10, 90, 1184, 20650, 450252, 11803526, 361892848, 12712357170, 503564718260, 22212233618542, 1079909444635848, 57379354040049002, 3308238701451609772, 205715613407117613270, 13724187813695296374752, 977841609869801208944482, 74108335568947966714172004
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2024

Keywords

Crossrefs

Cf. A161633, A377545, A377551.
Cf. A364980.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364980.
a(n) = 2 * n! * Sum_{k=0..n} k^(n-k) * binomial(2*n-k+2,k)/( (2*n-k+2)*(n-k)! ).

A377545 E.g.f. satisfies A(x) = 1/(1 - x * exp(x*A(x)))^3.

Original entry on oeis.org

1, 3, 18, 195, 3108, 65595, 1730538, 54891165, 2036187576, 86536398195, 4147191867630, 221314773837333, 13017260705093604, 836754118106509083, 58364080427471191506, 4390560359156841730605, 354356981533262814367728, 30543768949098926368973667, 2800395449868306713606542422
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2024

Keywords

Crossrefs

Cf. A161633, A377541, A377551.
Cf. A364981.

Programs

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364981.
a(n) = 3 * n! * Sum_{k=0..n} k^(n-k) * binomial(3*n-2*k+3,k)/( (3*n-2*k+3)*(n-k)! ).

A377550 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x*A(x)^4).

Original entry on oeis.org

1, 1, 4, 45, 772, 17865, 525966, 18794881, 790175128, 38221092657, 2091074167450, 127675964340441, 8606833626646740, 634928943628432921, 50878715440232312374, 4400937219238706030865, 408700742920092110904496, 40558224679468186878237153, 4283310197644529184427059378
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2024

Keywords

Crossrefs

Cf. A161633, A364980, A364981.
Cf. A377551, A377552.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*n-3*k+1, k)/((4*n-3*k+1)*(n-k)!));

Formula

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

A377630 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^4 ).

Original entry on oeis.org

1, 4, 52, 1212, 41512, 1889700, 107684664, 7384011796, 592485333472, 54488274328836, 5652345176418280, 653054114586249684, 83175314479016845584, 11578838832843098353732, 1749242011108507789948312, 285034599164755404426493140, 49833544890911336997795542464
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Links

Crossrefs

Cf. A161633, A377553, A377554.
Cf. A364989, A377633.

Programs

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

Formula

E.g.f. satisfies A(x) = (1 + x * A(x) * exp(x*A(x)))^4.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A364989.
a(n) = (n!/(n+1)) * Sum_{k=0..n} k^(n-k) * binomial(4*n+4,k)/(n-k)!.

A380043 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*exp(x*A(x)^3) )^(1/3).

Original entry on oeis.org

1, 1, 6, 73, 1364, 34585, 1110406, 43200535, 1975744856, 103892750209, 6176282882570, 409635957376591, 29988473838531748, 2402004132488328433, 208956515057627326094, 19619264794744128427495, 1977503574407863125008816, 212975277029523353673126529, 24408338689788753822318157330
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2025

Keywords

Crossrefs

Cf. A380039, A380041.
Cf. A161633, A380042.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x/(1+3*x*exp(x)))/x)^(1/3)))
  • PARI
    a(n) = n!*sum(k=0, n, 3^k*k^(n-k)*binomial(n+1/3, k)/(n-k)!)/(3*n+1);

Formula

E.g.f.: ( (1/x) * Series_Reversion(x/(1 + 3*x*exp(x))) )^(1/3).
a(n) = (n!/(3*n+1)) * Sum_{k=0..n} 3^k * k^(n-k) * binomial(n+1/3,k)/(n-k)!.
Previous Showing 21-30 of 34 results. Next

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

Content is available under The OEIS End-User License Agreement.