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 1-8 of 8 results.

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).

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

A365186 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)^2).

Original entry on oeis.org

1, 1, 6, 47, 428, 4241, 44407, 483358, 5414618, 62014112, 722870120, 8547768832, 102284029963, 1236274747490, 15070955944288, 185089043535730, 2287843817573898, 28440852786725695, 355345599519983962, 4459821165693379625, 56200963128262312342
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002295, A365184, A365185, A365187, A365188, A365189.
Cf. A365192.

Programs

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

Formula

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

A365185 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)).

Original entry on oeis.org

1, 1, 6, 46, 411, 3996, 41062, 438662, 4823133, 54221518, 620404859, 7201317005, 84590041441, 1003656037278, 12010861830069, 144804336388912, 1757106190680819, 21443109365898743, 263009775111233392, 3240530659303505547, 40088688455992604594
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002295, A365184, A365186, A365187, A365188, A365189.
Cf. A364748.

Programs

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

Formula

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

A365187 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)^3).

Original entry on oeis.org

1, 1, 6, 48, 446, 4511, 48218, 535800, 6127598, 71648868, 852668952, 10293847592, 125759270354, 1551872951050, 19314892116764, 242182938963024, 3056337851481678, 38790948190319404, 494825459824571528, 6340628082364678016, 81577931200018721464
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002295, A365184, A365185, A365186, A365188, A365189.
Cf. A365193.

Programs

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

Formula

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

A365188 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)^4).

Original entry on oeis.org

1, 1, 6, 49, 465, 4807, 52533, 596936, 6981798, 83497115, 1016367737, 12550853210, 156845913315, 1979870172453, 25207383853375, 323325558146400, 4174108907656633, 54195445136831670, 707225283913589280, 9270735916525207605, 122020617365557674605
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A002295, A365184, A365185, A365186, A365187, A365189.
Cf. A243667.

Programs

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

Formula

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

A376328 G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x)))^5.

Original entry on oeis.org

1, 5, 40, 380, 3970, 44051, 509575, 6077435, 74194780, 922644310, 11646083631, 148827827450, 1921724362880, 25034267112600, 328614891689845, 4342322118727300, 57715241768897445, 771087466276360970, 10349495416322497575, 139486475071720234920, 1886980259513934080860, 25613816043115261657425
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2024

Keywords

Links

Crossrefs

Cf. A143927, A365128, A376320.
Cf. A365189.

Programs

  • PARI
    a(n, s=1, t=5) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^2)^5)/x)

Formula

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).
G.f.: (1/x) * Series_Reversion( x / (1+x+x^2)^5 ).
G.f.: B(x)^5, where B(x) is the g.f. of A365189.
Showing 1-8 of 8 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.