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 14 results. Next

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).

A365184 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x).

Original entry on oeis.org

1, 1, 6, 45, 395, 3775, 38146, 400826, 4335455, 47951065, 539823620, 6165377836, 71261299056, 831990025420, 9797505040130, 116235417614900, 1387958781395535, 16668362761081560, 201190667288072005, 2439418470063468505, 29698136499328762445
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Links

Crossrefs

Cf. A002295, A365185, A365186, A365187, A365188, A365189.
Cf. A364475, A365178.
Cf. A002294, A349332.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(k,n-k)/(5*k+1) = Sum_{k=0..n} binomial(k,n-k) * A002294(k).

A365180 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x*A(x)).

Original entry on oeis.org

1, 1, 5, 31, 223, 1740, 14328, 122549, 1078197, 9695359, 88710199, 823247686, 7730244098, 73310150097, 701163085849, 6755544043969, 65506554804129, 638794412442172, 6260571309256152, 61632794482411367, 609197871548209907, 6043456939539775056
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002294, A365178, A365181, A365182, A365183.
Cf. A364747.

Programs

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

Formula

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

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).

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).

A371577 G.f. satisfies A(x) = ( 1 + x*A(x)^2 * (1 + x) )^2.

Original entry on oeis.org

1, 2, 11, 70, 505, 3910, 31772, 267280, 2307982, 20339946, 182207333, 1654250474, 15187764411, 140767293560, 1315349040350, 12377806027892, 117200381305538, 1115791797318548, 10674418686087377, 102563189093302366, 989321056200478417
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2024

Keywords

Crossrefs

Cf. A001629, A052705, A371576, A371578.
Cf. A365178.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(4*k+2,k) * binomial(k,n-k)/(2*k+1).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A365178.
Showing 1-10 of 14 results. Next

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

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