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.

Previous Showing 31-37 of 37 results.

A369631 Expansion of (1/x) * Series_Reversion( x * (1/(1+x^4) - x) ).

Original entry on oeis.org

1, 1, 2, 5, 15, 49, 168, 594, 2149, 7920, 29640, 112359, 430564, 1665197, 6491280, 25478886, 100611695, 399421439, 1593221090, 6382176160, 25664184349, 103560846454, 419215870860, 1701907025715, 6927658961599, 28268225980197, 115608889788304
Offset: 0

Views

Author

Seiichi Manyama, Jan 28 2024

Keywords

Links

Crossrefs

Cf. A007863, A199874, A369630.
Cf. A227035.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1/(1+x^4)-x))/x)
  • PARI
    a(n) = sum(k=0, n\4, binomial(2*n-4*k+1, k)*binomial(2*n-4*k, n-4*k))/(n+1);

Formula

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

A379023 Expansion of (1/x) * Series_Reversion( x * ((1 - x - x^2)/(1 + x))^3 ).

Original entry on oeis.org

1, 6, 57, 653, 8277, 111780, 1576671, 22955298, 342377304, 5204438258, 80334470136, 1255798641861, 19840021268937, 316286673287724, 5081503084814883, 82193597974971157, 1337397202150986387, 21875767255039745856, 359499909751084059372, 5932767953991599086905
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2024

Keywords

Links

Crossrefs

Cf. A007863, A379021, A379024.
Cf. A239107, A379025.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379025(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x*A(x)^(4/3)) )^3.
(3) A(x) = B(x)^3 where B(x) is the g.f. of A239107.
a(n) = (1/(n+1)) * [x^n] ( (1 + x)/(1 - x - x^2) )^(3*(n+1)).
a(n) = 3 * Sum_{k=0..n} binomial(3*n+k+3,k) * binomial(3*n+k+3,n-k)/(3*n+k+3) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(3*n+k+3,n-k).

A011274 Triangle of numbers of hybrid rooted trees (divided by Fibonacci numbers).

Original entry on oeis.org

1, 2, 1, 7, 4, 1, 31, 18, 6, 1, 154, 90, 33, 8, 1, 820, 481, 185, 52, 10, 1, 4575, 2690, 1065, 324, 75, 12, 1, 26398, 15547, 6276, 2006, 515, 102, 14, 1, 156233, 92124, 37711, 12468, 3420, 766, 133, 16, 1, 943174, 556664, 230277, 78030, 22412, 5439, 1085, 168, 18, 1
Offset: 1

Views

Author

Jean Pallo (pallo(AT)u-bourgogne.fr)

Keywords

Comments

Triangle T(n,k) = [x^(n-k)] A(x)^k where A(x) is the o.g.f. of A007863. - Vladimir Kruchinin, Mar 17 2011
Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A007863. - Philippe Deléham, Feb 03 2014

Examples

			     1
     2    1
     7    4    1
    31   18    6   1
   154   90   33   8  1
   820  481  185  52 10  1
  4575 2690 1065 324 75 12 1
Production matrix is:
   2   1
   3   2   1
   5   3   2   1
   8   5   3   2   1
  13   8   5   3   2   1
  21  13   8   5   3   2   1
  34  21  13   8   5   3   2   1
  55  34  21  13   8   5   3   2   1
  89  55  34  21  13   8   5   3   2   1
  ... - _Philippe Deléham_, Feb 03 2014

Links

Crossrefs

Cf. A000045, A011270, A011272.

Programs

  • Maple
    A011274 := proc(n,k) k/n*add( binomial(i+n-1,n-1)*binomial(i+n,n-k-i),i=0..n-k) ; end proc: # R. J. Mathar, Mar 21 2011
  • Mathematica
    t[n_, k_] := k/n*Binomial[n, k]*HypergeometricPFQ[ {k-n, n, n+1}, {1/2 + k/2, 1+k/2}, -1/4]; Flatten[ Table[ t[n, k], {n, 1, 10}, {k, 1, n}]] (* Jean-François Alcover, Dec 02 2011, after Vladimir Kruchinin *)
  • Maxima
    A011274(n,k):= k/n*sum(binomial(i+n-1,n-1)*binomial(i+n,n-k-i), i,0,n-k); /* Vladimir Kruchinin, Mar 17 2011 */

Formula

T(n,k) = (k/n) *Sum_{i=0..n-k} binomial(i+n-1,n-1)*binomial(i+n,n-k-i). - Vladimir Kruchinin, Mar 17 2011
(r/(m*n+r))*T((m+1)*n+r,m*n+r) = Sum_{k=1..n} k*T((m+1)*n-k,m*n)*T(r+k,r)/n. - Vladimir Kruchinin, Mar 17 2011
T(n,m) = (m/n)*Sum_{k=1..n-m+1} k*A007863(k-1)*T(n-k,m-1), 1 < m <= n. - Vladimir Kruchinin, Mar 17 2011

A212694 Number of 2-colored Dyck n-paths with up steps (U, u), down steps (D, d), and avoiding UU and DD.

Original entry on oeis.org

1, 4, 25, 197, 1745, 16580, 165115, 1700809, 17971466, 193710087, 2121585340, 23543198588, 264138223362, 2991130956918, 34143543312267, 392458689992396, 4538574332686469, 52768896995910303, 616471818881678085, 7232838546289017796, 85188401983572928395
Offset: 0

Views

Author

Alois P. Heinz, May 23 2012

Keywords

Comments

Upper case letters denote one color and lower case letters the other.

Examples

			a(1) = 4: ud, Ud, uD, UD.
a(2) = 25: uudd, Uudd, uUdd, uuDd, UuDd, uUDd, udud, Udud, uDud, UDud, udUd, UdUd, uDUd, UDUd, uudD, UudD, uUdD, uduD, UduD, uDuD, UDuD, udUD, UdUD, uDUD, UDUD.

Links

Crossrefs

Cf. A000108, A007863, A151403.

Programs

  • Maple
    b:= proc(x, y, t) option remember; `if`(x=0, 1,
      `if`(y<1  , 0, b(x-1, y-1, 0)+`if`(t=1, 0, b(x-1, y-1, 1)))+
      `if`(x b(2*n, 0$2):
seq(a(n), n=0..30);
  • Mathematica
    b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, If[y < 1, 0, b[x - 1, y - 1, 0] + If[t == 1, 0, b[x - 1, y - 1, 1]]] + If[x < y + 2, 0, b[x - 1, y + 1, 0] + If[t == 2, 0, b[x - 1, y + 1, 2]]]];
a[n_] := b[2n, 0, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Formula

Recurrence: 5*n*(n+1)*(n+2)*(11856*n^6 - 462048*n^5 + 6376819*n^4 - 42412433*n^3 + 147659510*n^2 - 260432089*n + 184927095)*a(n) = n*(n+1)*(1683552*n^7 - 66428880*n^6 + 937628962*n^5 - 6466755025*n^4 + 23922419618*n^3 - 47274340850*n^2 + 44490285903*n - 13108829940)*a(n-1) - n*(14310192*n^8 - 585624672*n^7 + 8802419365*n^6 - 66744441981*n^5 + 284660409448*n^4 - 703230360993*n^3 + 976943147665*n^2 - 682900257024*n + 175305383460)*a(n-2) + 2*(17238624*n^9 - 746332752*n^8 + 12295273466*n^7 - 105941663163*n^6 + 537013083761*n^5 - 1677467513328*n^4 + 3243096592679*n^3 - 3743377415442*n^2 + 2332897216785*n - 592736810400)*a(n-3) - (37974768*n^9 - 1728832752*n^8 + 30714335273*n^7 - 291055104422*n^6 + 1652510618897*n^5 - 5890634883203*n^4 + 13267453974662*n^3 - 18291224811263*n^2 + 14073816074160*n - 4638445144200)*a(n-4) + (23308896*n^9 - 1108835760*n^8 + 20950004098*n^7 - 213362099351*n^6 + 1311302140489*n^5 - 5085585201440*n^4 + 12508042515937*n^3 - 18889708965719*n^2 + 15984167161110*n - 5834960787600)*a(n-5) - (8927568*n^9 - 442319616*n^8 + 8817227331*n^7 - 95321285525*n^6 + 623567997102*n^5 - 2575734547859*n^4 + 6742249614729*n^3 - 10822975984520*n^2 + 9730758446550*n - 3782772932400)*a(n-6) + 4*(2*n - 13)*(248976*n^8 - 11244288*n^7 + 197289135*n^6 - 1812121625*n^5 + 9661474889*n^4 - 30841990357*n^3 + 57959845045*n^2 - 59304520365*n + 25796647800)*a(n-7) - 4*(n-7)*(2*n - 15)*(2*n - 13)*(11856*n^6 - 390912*n^5 + 4244419*n^4 - 21288517*n^3 + 54240485*n^2 - 69082196*n + 35668710)*a(n-8). - Vaclav Kotesovec, Jul 16 2014
Limit n->infinity a(n)^(1/n) = (13+3*sqrt(17))/2 = 12.68465843842649... . - Vaclav Kotesovec, Jul 16 2014

A234938 Coefficients of Hilbert series for the suboperad of bicolored noncrossing configurations generated by a fully colored triangle and a fully uncolored triangle.

Original entry on oeis.org

1, 2, 8, 40, 216, 1246, 7516, 46838, 299200, 1948804, 12893780, 86415940, 585461380, 4003022222, 27587072156, 191426864328, 1336331235624, 9378578814890, 66133103587412, 468323884345060, 3329180643569660, 23748479467116032, 169944228206075568, 1219639212041064130
Offset: 1

Views

Author

N. J. A. Sloane, Jan 04 2014

Keywords

Links

Crossrefs

Cf. A234939, A052701, A007863, A006013, A006318.

Programs

  • Mathematica
    Rest@CoefficientList[Root[Function[{f}, 4t-2t^2-t^3+t^4 + (-4+4t-t^2+2t^3)f + (6+t)f^2 + (1-2t)f^3 - f^4], 2] + O[t]^25, t] (* Andrey Zabolotskiy, Feb 02 2025 *)

Formula

G.f. A(t) satisfies 4t-2t^2-t^3+t^4 + (-4+4t-t^2+2t^3)*A(t) + (6+t)*A(t)^2 + (1-2t)*A(t)^3 - A(t)^4 = 0 [Chapoton & Giraudo, Proposition 3.5]. - Andrey Zabolotskiy, Feb 02 2025

Extensions

Terms a(9) onwards added and name clarified by Andrey Zabolotskiy, Feb 02 2025

A192204 G.f.: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^4*A(x)^k) * x^n/n ).

Original entry on oeis.org

1, 2, 13, 109, 1099, 12283, 147620, 1869346, 24633344, 334916467, 4669887745, 66481991644, 963096090267, 14160279233964, 210870471771803, 3175275874056722, 48281516978747396, 740504452581897112, 11444972742343813815
Offset: 0

Views

Author

Paul D. Hanna, Jun 25 2011

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 13*x^2 + 109*x^3 + 1099*x^4 + 12283*x^5 +...
which satisfies:
log(A(x)) = (1 + A(x))*x + (1 + 16*A(x) + A(x)^2)*x^2/2 + (1 + 81*A(x) + 81*A(x)^2 + A(x)^3)*x^3/3 + (1 + 256*A(x) + 1296*A(x)^2 + 256*A(x)^3 + A(x)^4)*x^4/4 +...

Crossrefs

Cf. variants: A007863, A192131.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^4*(A+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

A379139 G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^(1/3)) * (1 + x*A(x)^(2/3)) )^3.

Original entry on oeis.org

1, 6, 33, 185, 1065, 6276, 37711, 230277, 1425180, 8920915, 56382321, 359325561, 2306603557, 14900834070, 96801107625, 631995206707, 4144614844047, 27289670546697, 180339237891360, 1195684324845420, 7951560286540908, 53025939926690233, 354509890878236583
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2024

Keywords

Crossrefs

Cf. A003517, A007863.

Programs

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

Formula

G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A007863.
a(n) = 3 * Sum_{k=0..n} binomial(n+k+3,k) * binomial(n+k+3,n-k)/(n+k+3).
Previous Showing 31-37 of 37 results.

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

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