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 1-4 of 4 results.

A367233 G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - x*A(x))^3.

Original entry on oeis.org

1, 1, 6, 39, 284, 2223, 18267, 155445, 1358073, 12111306, 109802183, 1009001571, 9376972698, 87978198364, 832223905371, 7928413841673, 76002832317437, 732578811761670, 7095717550127526, 69029297500888522, 674181392461483212, 6607910786529613248
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2023

Keywords

Links

Crossrefs

Cf. A161797, A365113, A365150.
Cf. A108447, A367232, A367235.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).

A367235 G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - x*A(x))^4.

Original entry on oeis.org

1, 1, 7, 50, 399, 3422, 30798, 286974, 2744947, 26798010, 265945022, 2674970684, 27209385886, 279412999031, 2892787737002, 30161921520976, 316440334960563, 3338105334701396, 35385133077851602, 376732207920371784, 4026682585718602014
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2023

Keywords

Links

Crossrefs

Cf. A321798, A365114, A367234.
Cf. A108447, A367232, A367233.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).

A367259 G.f. satisfies A(x) = 1 + x*A(x)^3 * (1 + x*A(x))^2.

Original entry on oeis.org

1, 1, 5, 27, 169, 1138, 8061, 59188, 446455, 3438863, 26935372, 213883631, 1717852129, 13931065117, 113913095218, 938154381748, 7774936633411, 64791892224825, 542598513709481, 4564001359135661, 38541714429405304, 326640923339410701
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2023

Keywords

Crossrefs

Cf. A007477, A161634, A214372.
Cf. A367232.

Programs

  • Maple
    A367259 := proc(n)
    add(binomial(3*k+(n-k)+1,k) * binomial(2*k,n-k) / (3*k+(n-k)+1),k=0..n) ;
end proc:
seq(A367259(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    a(n, s=2, t=3, u=1) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).
D-finite with recurrence 8*n*(26345215448853860010445423574*n -20961990363613887876514780359) *(2*n+1)*(2*n-1) *(n+1)*a(n) +8*n*(2*n-1)* (52690430897707720020890847148*n^3 -8177454564962587489822763646077*n^2 +19278991143331529980160099092658*n-11257584930903257675329694643457) *a(n-1) +2*(-268110383402413819740981254825038*n^5 +2815977437639263120434136294300085*n^4 -10349136726006717489413692948200650*n^3 +17659039091779726381787370980047525*n^2 -14385155927699861644653059971375422*n +4528097093401255127907905744957880) *a(n-2) +2*(2433809541139490470204589489378644*n^5 -29695021140710269817720089645612595*n^4 +147295722233051282998786410783007430*n^3 -369735985683645289967183608338045205*n^2 +467786753736867474630837654962591406*n -237792800129696483300250545739991320) *a(n-3) +2*(-2771166843885660051994398777044296*n^5 +70669385063622159693270493531099173*n^4 -615983650096141972053534317661369592*n^3 +2483715780351994976504831765723882733*n^2 -4785455586973998561063713309602358866*n +3584098048545781487176463022333484200) *a(n-4) +(-9719685405660460345432742140418255*n^5 +101488259196839193588678566387929584*n^4 +90729575616085725241944658061815579*n^3 -4613811089886954307541928620224211376*n^2 +19004407111946953332012410754931442164*n -24034005967022354223275806054437127680) *a(n-5) +5*(4577999937824616490204866357596875*n^5 -139866352876176382080407833814299250*n^4 +1615537655758910403049946493111918725*n^3 -8980705919938192198141139371077714070*n^2 +24274376174445463863335689528941329496*n -25684691566228873512769902557616240960) *a(n-6) +30*(1336493817891495200869338759185*n -5552932550849126027962496889033) *(5*n-32)*(5*n-26)*(5*n-29)*(5*n-28)*a(n-7)=0. - R. J. Mathar, Dec 04 2023

A367280 G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - x*A(x)^3)^2.

Original entry on oeis.org

1, 1, 5, 33, 251, 2073, 18069, 163600, 1523731, 14504988, 140499307, 1380322749, 13721269995, 137758098052, 1394840743638, 14227181658075, 146048314214619, 1507739540085350, 15643456882376418, 163036276218805231, 1706021256401103673
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2023

Keywords

Crossrefs

Cf. A002294, A366176, A367232, A367238.
Cf. A002293, A367242.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).
Showing 1-4 of 4 results.

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

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