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 11 results. Next

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).

A255673 Coefficients of A(x), which satisfies: A(x) = 1 + x*A(x)^3 + x^2*A(x)^6.

Original entry on oeis.org

1, 1, 4, 21, 127, 833, 5763, 41401, 305877, 2309385, 17739561, 138197876, 1089276972, 8670856834, 69606939717, 562879492551, 4580890678781, 37490975387565, 308369889858450, 2547741413147700, 21133987935358776, 175947462569886786, 1469656053534121804
Offset: 0

Views

Author

Werner Schulte, Jul 10 2015

Keywords

Comments

This sequence is the next after A001006 and A006605.

Examples

			A(x) = 1 + x + 4*x^2 + 21*x^3 + 127*x^4 + 833*x^5 + 5763*x^6 ...

Links

Crossrefs

Cf. A001006, A006605.
Cf. A365183, A365189.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(1-A+x*A^3+x^2*A^6, A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 15 2015
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 1, 9*(((3*n-1))*
     (2*n-1)*(3*n-2)*(9063*n^4-18126*n^3+8403*n^2+660*n-280)*a(n-1)
     +(27*(n-1))*(3*n-1)*(3*n-4)*(3*n-2)*(3*n-5)*(57*n^2-2)*a(n-2))
      /((5*(5*n+2))*(5*n-1)*(5*n+1)*(5*n-2)*n*(57*n^2-114*n+55)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 16 2015
  • Mathematica
    m = 30; A[_] = 0;
Do[A[x_] = 1 + x A[x]^3 + x^2 A[x]^6 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 04 2019 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(n-k, k)*binomial(3*n+1, n-k))/(3*n+1); \\ Seiichi Manyama, Sep 02 2023

Formula

a(n+1) = Sum_{j=0..3*n+4} binomial(j,2*j-5*n-7) * binomial(3*n+4,j) / (3*n+4). (conjectured). [Vladimir Kruchinin, Mar 09 2013]
a(n) = 1/(3*n+1) * Sum_{k=0..n} (-1)^k * binomial(3*n+1, k) * binomial(6*n+2-2*k, n-k). (conjectured)
G.f. A(x) satisfies A(x) = G(x*A(x)), where G is g.f. of A006605.
G.f. A(x) satisfies A(x) = H(x*A(x)^2), where H is g.f. of A001006.
From Peter Bala, Jul 27 2023: (Start)
Define b(n) = [x^n] (1 + x + x^2)^(3*n). Then A(x)^3 = exp(Sum_{n >= 1} b(n)*x^n/n).
A(x^3) = (1/x) * series reversion of x/(1 + x^3 + x^6) = 1 + x^3 + 4*x^6 + 21*x^9 + 127*x^12 + .... (End)
a(n) = (1/(3*n+1)) * Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(3*n+1,n-k). - Seiichi Manyama, Sep 02 2023

Extensions

More terms from Alois P. Heinz, Jul 15 2015

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).

A365189 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)^5).

Original entry on oeis.org

1, 1, 6, 50, 485, 5130, 57391, 667777, 7999095, 97986680, 1221813880, 15456556791, 197887386913, 2559189842240, 33383097891135, 438714241508615, 5803049210371375, 77199163872173757, 1032215519193531310, 13864180990526161995, 186975433988014039830
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Links

Crossrefs

Cf. A002295, A365184, A365185, A365186, A365187, A365188.
Cf. A006605, A255673, A365183.

Programs

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

Formula

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

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).
Showing 1-10 of 11 results. Next

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