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 1-5 of 5 results.

A073157 Number of Schroeder n-paths containing no FFs.

Original entry on oeis.org

1, 2, 5, 18, 70, 293, 1283, 5808, 26960, 127628, 613814, 2990681, 14730713, 73229291, 366936231, 1851352820, 9397497758, 47957377934, 245903408244, 1266266092112, 6545667052320, 33954266444498, 176689391245146
Offset: 0

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Author

Paul D. Hanna, Jul 29 2002

Keywords

Comments

Number of Schroeder n-paths containing no FFs. A Schroeder n-path (A006318) consists of steps U=(1,1),F=(2,0),D=(1,-1) starting at (0,0), ending at (2n,0), and never going below the x-axis. Example: a(2)=5 counts UFD, UUDD, UDF, FUD, UDUD. - David Callan, Aug 23 2011

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 70*x^4 + 293*x^5 + 1283*x^6 + ...

Links

Crossrefs

Leftmost column of triangle A073154 (was previous name).
Cf. A000108, A073155, A073156, A073153, A143330, A198953, A215715, A216359, A364335, A364371.

Programs

  • GAP
    List([0..25],n->Sum([0..n],i->Binomial(2*i+1,i)*Binomial(2*i+1,n-i)/(2*i+1))); # Muniru A Asiru, Oct 11 2018
  • Maple
    a:=n->add(binomial(2*i+1,i)*binomial(2*i+1,n-i)/(2*i+1),i=0..n): seq(a(n),n=0..25); # Muniru A Asiru, Oct 11 2018
  • Mathematica
    Table[Sum[Binomial[2*i + 1, i]*Binomial[2*i + 1, n - i]/(2*i + 1), {i, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 11 2018 *)
  • Maxima
    a(n):=sum((sum((binomial(2*k+2,j-k)*binomial(2*k,k)/(k+1)),k,0,j))*(-1)^(n-j),j,0,n); /* Vladimir Kruchinin, Mar 13 2016 */
  • PARI
    {a(n)=local(A=1); for(i=0,n-1,A=(1+x)*(1+x*(A+x*O(x^n))^2));polcoeff(A,n)} /* Paul D. Hanna, Mar 03 2008 */

Formula

A073155(n+1) = Sum_{k=0..n} a(k)*a(n-k), that is, convolution yields sequence A073155 minus the 0th term.
G.f.: A(x) = (1 - sqrt(1 - 4*x*(1+x)^2))/(2*x*(1+x)) satisfies A(x) = (1+x)*(1 + x*A(x)^2);
G.f.: A(x) = (1+x)*C(x*(1+x)^2) where C(x) is the Catalan g.f. of A000108. - Paul D. Hanna, Mar 03 2008
a(n) = Sum_{j=0..n}((Sum_{k=0..j}((binomial(2*k+2,j-k)*C(k))))*(-1)^(n-j)), where C(k) = A000108(k). - Vladimir Kruchinin, Mar 13 2016
a(n) = Sum_{i=0..n} C(2*i+1,i)*C(2*i+1,n-i)/(2*i+1). - Vladimir Kruchinin, Oct 11 2018
Recurrence: (n+1)*a(n) = 3*(n-1)*a(n-1) + 6*(2*n - 3)*a(n-2) + 6*(2*n - 5)*a(n-3) + 2*(2*n - 7)*a(n-4). - Vaclav Kotesovec, Oct 11 2018
From Peter Bala, Aug 25 2024: (Start)
(1/x) * series_reversion(x/A(x)) = 1 + 2*x + 9*x^2 + 56*x^3 + 400*x^4 + 3095*x^5 + 25240*x^6 + ... is the g.f. of A198953.
(1/x) * series_reversion(x*A(-x)) = 1 + 2*x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 + ... = G(x) + x, where G(x) = (1 - x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x) is the g.f. of A143330. (End)
Define a sequence operator R: {u(n): n >= 0} -> {v(n): n >= 0} by Sum_{n >= 0} v(n)*x^n = (1/x) * series_reversion(x/Sum_{n >= 0} u(n)*x^n). Then R({a(n)}) = A198953, R^2({a(n)}) = A215715 and R^3({a(n)}) = A364335. Cf. A216359. - Peter Bala, Sep 13 2024

Extensions

More terms from Paul D. Hanna, Mar 03 2008
New name using a comment from David Callan, Peter Luschny, Oct 14 2018

A364337 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^4).

Original entry on oeis.org

1, 2, 9, 68, 580, 5406, 53270, 545844, 5757332, 62094217, 681653493, 7591431752, 85558696024, 974024788280, 11184192097016, 129378232148016, 1506363564912368, 17639001584452320, 207593804132718948, 2454236122156830254, 29132714097692056954, 347086786035103983446
Offset: 0

Views

Author

Seiichi Manyama, Jul 19 2023

Keywords

Links

Crossrefs

Cf. A073157, A364336, A364338, A364339.
Cf. A215623, A215715, A234461, A239107.

Programs

  • Mathematica
    terms = 22; A[] = 0; Do[A[x] = (1+x)(1+x*A[x]^4) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Mar 24 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(4*k+1, k)*binomial(4*k+1, n-k)/(4*k+1));

Formula

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

A364331 G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^5).

Original entry on oeis.org

1, 2, 15, 163, 2070, 28698, 421015, 6425644, 100977137, 1622885389, 26551709946, 440744175801, 7404449354076, 125657625548824, 2150963575012295, 37094953102567208, 643904274979347286, 11241232087809137759, 197247501440314516840, 3476787208220672891388, 61533794803235280779261
Offset: 0

Views

Author

Seiichi Manyama, Jul 18 2023

Keywords

Crossrefs

Cf. A007863, A069271, A073157, A215654, A215715, A364333.
Cf. A215623, A215624, A239108, A364335, A364338.
Cf. A200719.

Programs

  • Maple
    A364331 := proc(n)
    add( binomial(2*n+3*k+1,k) * binomial(2*n+3*k+1,n-k)/(2*n+3*k+1),k=0..n) ;
end proc:
seq(A364331(n),n=0..70); # R. J. Mathar, Jul 25 2023
  • PARI
    a(n) = sum(k=0, n, binomial(2*n+3*k+1, k)*binomial(2*n+3*k+1, n-k)/(2*n+3*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(2*n+3*k+1,k) * binomial(2*n+3*k+1,n-k) / (2*n+3*k+1).
x/series_reversion(x*A(x)) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + ..., the g.f. of A215623. - Peter Bala, Sep 08 2024

A364333 G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^6).

Original entry on oeis.org

1, 2, 17, 216, 3224, 52640, 910452, 16392140, 303996224, 5767278431, 111401778266, 2183535060362, 43319505976084, 868220464851417, 17552981176788200, 357544690982030744, 7330803752675100908, 151172599088871911072, 3133367418601958989295, 65242183918761533467216
Offset: 0

Views

Author

Seiichi Manyama, Jul 18 2023

Keywords

Crossrefs

Cf. A007863, A069271, A073157, A215654, A215715, A364331.
Cf. A239109, A364339, A364340.
Cf. A200718.

Programs

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

Formula

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

A379279 G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^2) * (1 + x*A(x)) )^2.

Original entry on oeis.org

1, 4, 30, 288, 3125, 36490, 447478, 5683186, 74105002, 986302778, 13344661479, 182998935930, 2537838036761, 35530970858236, 501523116910044, 7129275916213606, 101973703002773268, 1466574750062589956, 21194869324964207133, 307642575576365729486, 4482940969372057898247
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2024

Keywords

Crossrefs

Cf. A371675, A379251, A379282.
Cf. A073155, A379280.
Cf. A215715.

Programs

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

Formula

G.f.: B(x)^2 where B(x) is the g.f. of A215715.
a(n) = Sum_{k=0..n} binomial(2*n+2*k+2,k) * binomial(2*n+2*k+2,n-k)/(n+k+1).
Showing 1-5 of 5 results.

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