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.

A365178 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x).

Original entry on oeis.org

1, 1, 5, 30, 210, 1595, 12791, 106574, 913562, 8004861, 71375653, 645536234, 5907683486, 54605672300, 509043322720, 4780441915832, 45182744331388, 429472919087158, 4102806757542542, 39370967793387086, 379335734835510622, 3668220243145708341
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Links

Crossrefs

Cf. A002294, A365180, A365181, A365182, A365183.
Cf. A364475, A365184.
Cf. A002293, A364987.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(k,n-k)/(4*k+1) = Sum_{k=0..n} binomial(k,n-k) * A002293(k).

A365183 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x*A(x)^4).

Original entry on oeis.org

1, 1, 5, 34, 268, 2299, 20838, 196326, 1903524, 18868861, 190356231, 1948055058, 20173907384, 211020478270, 2226243632838, 23660868061422, 253099278807684, 2722819049879436, 29439894433161189, 319749417998303470, 3486914150183526920
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Links

Crossrefs

Cf. A002294, A365178, A365180, A365181, A365182.
Cf. A006605, A255673, A365189.
Cf. A364989.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n-k, k)*binomial(4*n+1, n-k))/(4*n+1);

Formula

a(n) = (1/(4*n+1)) * Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(4*n+1,n-k).

A365182 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x*A(x)^3).

Original entry on oeis.org

1, 1, 5, 33, 252, 2091, 18319, 166750, 1561599, 14948572, 145615404, 1438752770, 14384289530, 145248707646, 1479212551278, 15175516654760, 156691764630780, 1627069871618145, 16980373299730925, 178006989972532900, 1873607777794186000
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002294, A365178, A365180, A365181, A365183.
Cf. A365177.

Programs

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

Formula

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

A365181 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x*A(x)^2).

Original entry on oeis.org

1, 1, 5, 32, 237, 1905, 16160, 142392, 1290613, 11955947, 112697701, 1077438356, 10422562156, 101827196684, 1003312506776, 9958506719664, 99479743121349, 999370184665407, 10090067735619023, 102330789530653912, 1041997707624103589, 10648963961114066129
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002294, A365178, A365180, A365182, A365183.

Programs

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

Formula

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

A367041 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^4.

Original entry on oeis.org

1, 1, 5, 26, 168, 1195, 8988, 70318, 566388, 4665221, 39113732, 332691758, 2863778072, 24900264326, 218372530380, 1929363592870, 17157018725000, 153442147343648, 1379250344938676, 12453816724761706, 112907775890596400, 1027394297869071687
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A176697, A367040.
Cf. A365178, A365180, A365181, A365182, A365183, A367048, A367049, A367050.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(3*(n-2*k)+1, k)*binomial(4*(n-2*k), n-2*k)/(3*(n-2*k)+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*(n-2*k)+1,k) * binomial(4*(n-2*k),n-2*k)/(3*(n-2*k)+1).

A200754 G.f. satisfies A(x) = 1 + x*A(x)^4 - x^2*A(x)^5.

Original entry on oeis.org

1, 1, 3, 13, 67, 380, 2288, 14351, 92737, 613063, 4126289, 28179766, 194780822, 1360053081, 9578997279, 67971291791, 485464864401, 3487203531460, 25176899072984, 182598098616625, 1329716528758651, 9718954060263384, 71273846758123552, 524279847227139350
Offset: 0

Views

Author

Paul D. Hanna, Nov 21 2011

Keywords

Comments

Compare to the g.f. G(x) for the ternary tree numbers (A001764): G(x) = 1 + x*G(x)^4 - x^2*G(x)^6 = 1 + x*G(x)^3.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 67*x^4 + 380*x^5 + 2288*x^6 +...
Related expansions:
A(x)^4 = 1 + 4*x + 18*x^2 + 92*x^3 + 515*x^4 + 3068*x^5 + 19092*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 780*x^4 + 4741*x^5 + 29915*x^6 +...
where a(2) = 4 - 1; a(3) = 18 - 5; a(4) = 92 - 25; a(5) = 515 - 135; ...

Links

Crossrefs

Cf. A200753, A200755, A365180.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*A^4-x^2*A^5+x*O(x^n));polcoeff(A,n)}
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n+3*k+1, k)*binomial(k, n-k)/(n+3*k+1)); \\ Seiichi Manyama, Nov 01 2023

Formula

Recurrence: 3*n*(n+1)*(3*n - 1)*(3*n + 1)*(1280*n^7 - 17280*n^6 + 94466*n^5 - 266799*n^4 + 407516*n^3 - 311946*n^2 + 81648*n + 12420)*a(n) = n*(281600*n^10 - 3942400*n^9 + 22465080*n^8 - 65757900*n^7 + 98435187*n^6 - 46293273*n^5 - 73831865*n^4 + 131625925*n^3 - 85490202*n^2 + 25470288*n - 2915460)*a(n-1) + (263680*n^11 - 4087040*n^10 + 26682396*n^9 - 96110406*n^8 + 210443037*n^7 - 293037231*n^6 + 270493529*n^5 - 181862299*n^4 + 103208358*n^3 - 47593224*n^2 + 12881700*n - 1360800)*a(n-2) - 5*(640000*n^11 - 10880000*n^10 + 78609000*n^9 - 311686500*n^8 + 721945299*n^7 - 918913929*n^6 + 367548335*n^5 + 642045653*n^4 - 1081692258*n^3 + 697174344*n^2 - 208955268*n + 24222240)*a(n-3) + 5*(5*n - 18)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(1280*n^7 - 8320*n^6 + 17666*n^5 - 8869*n^4 - 15820*n^3 + 22148*n^2 - 9282*n + 1305)*a(n-4). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ s*sqrt((2*r*s-1) / (2*Pi*(5*r*s-3))) / (2*n^(3/2)*r^n), where r = 0.1272568969777848138753091632571986265610307654216... and s = 1.358291097397172238669759690645074441686961930838... are roots of the system of equations s + r^2*s^5 = 1 + r*s^4, 1 + 5*r^2*s^4 = 4*r*s^3. - Vaclav Kotesovec, Nov 18 2017
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+3*k+1,k) * binomial(k,n-k)/(n+3*k+1). - Seiichi Manyama, Nov 01 2023

A367048 G.f. satisfies A(x) = 1 + x*A(x)^4 + x^2*A(x).

Original entry on oeis.org

1, 1, 5, 27, 177, 1270, 9645, 76206, 619913, 5156959, 43667985, 375140383, 3261467573, 28641957520, 253702185717, 2263964868768, 20334261430769, 183680693283325, 1667613040080061, 15208587941854251, 139266058402655669, 1279953660931370623
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A365178, A365180, A365181, A365182, A365183, A367041, A367049, A367050.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(3*n-5*k+1, k)*binomial(4*n-7*k, n-2*k)/(3*n-5*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-5*k+1,k) * binomial(4*n-7*k,n-2*k)/(3*n-5*k+1).

A367049 G.f. satisfies A(x) = 1 + x*A(x)^4 + x^2*A(x)^2.

Original entry on oeis.org

1, 1, 5, 28, 187, 1361, 10479, 83914, 691738, 5830903, 50028259, 435454040, 3835732631, 34128555184, 306276957665, 2769050552948, 25197515469820, 230599623819217, 2121066298440282, 19597929365099640, 181814132152022195, 1692920612932871541
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A365178, A365180, A365181, A365182, A365183, A367041, A367048, A367050.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(3*n-4*k+1, k)*binomial(4*n-6*k, n-2*k)/(3*n-4*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-4*k+1,k) * binomial(4*n-6*k,n-2*k)/(3*n-4*k+1).

A367050 G.f. satisfies A(x) = 1 + x*A(x)^4 + x^2*A(x)^3.

Original entry on oeis.org

1, 1, 5, 29, 198, 1469, 11518, 93875, 787392, 6752175, 58929541, 521718814, 4674070602, 42296077935, 386027716280, 3549332631052, 32845586854208, 305685481682970, 2859315003009776, 26866125820982711, 253457922829307765, 2399910588283502630
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A365178, A365180, A365181, A365182, A365183, A367041, A367048, A367049.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(3*n-3*k+1, k)*binomial(4*n-5*k, n-2*k)/(3*n-3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-3*k+1,k) * binomial(4*n-5*k,n-2*k)/(3*n-3*k+1).
Showing 1-9 of 9 results.

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

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