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 1-9 of 9 results.

A248382 Number of n-strand braids of length at most 2 in the dual monoid B_n^{+*}.

Original entry on oeis.org

1, 4, 83, 556, 11124, 266944
Offset: 1

Views

Author

N. J. A. Sloane, Oct 15 2014

Keywords

Links

Crossrefs

Cf. A000108, A168344, A248382-A248386.

A248386 Number of n-strand braids of length at most 6 in the dual monoid B_n^{+*}.

Original entry on oeis.org

1, 8, 1515, 334632
Offset: 1

Views

Author

N. J. A. Sloane, Oct 15 2014

Keywords

Links

Crossrefs

Cf. A000108, A168344, A248382-A248386.

A168450 G.f. A(x) satisfies: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = g.f. of A004304, where A004304(n) is the number of planar tree-rooted maps with n edges.

Original entry on oeis.org

1, 2, 6, 26, 148, 1012, 7824, 65886, 590452, 5546972, 54070432, 542937320, 5586265280, 58659600352, 626702981084, 6795682231830, 74645847739012, 829257675740724, 9304974123394272, 105343378754088424
Offset: 0

Views

Author

Paul D. Hanna, Nov 26 2009

Keywords

Examples

			G.f. A(x) = 1 + 2*x + 6*x^2 + 26*x^3 + 148*x^4 + 1012*x^5 + 7824*x^6 +...
A(x) satisfies: A(x*F(x)) = F(x) = g.f. of A005568:
F(x) = 1 + 2*x + 10*x^2 + 70*x^3 + 588*x^4 + 5544*x^5 + 56628*x^6 +...+ A000108(n)*A000108(n+1)*x^n +...
A(x) satisfies: A(x/G(x)) = G(x) = g.f. of A004304:
G(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 28*x^4 + 160*x^5 + 1036*x^6 +...

Links

Crossrefs

Cf. A004304, A005568, A000108, variant: A168344.

Programs

  • PARI
    {a(n)=local(C_2=vector(n+1,m,(binomial(2*m-2,m-1)/m)*(binomial(2*m,m)/(m+1))));polcoeff((x/serreverse(x*Ser(C_2))),n)}

Formula

G.f.: A(x) = F(x/A(x)) where A(x*F(x)) = F(x) = g.f. of A005568, where A005568(n) is the product of two successive Catalan numbers C(n)*C(n+1).
G.f.: A(x) = x/Series_Reversion(x*F(x)) where F(x) = g.f. of A005568.
G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) = g.f. of A004304.

A168357 Self-convolution of A006664, which is the number of irreducible systems of meanders.

Original entry on oeis.org

1, 2, 5, 20, 112, 768, 5984, 50856, 460180, 4366076, 42988488, 436066232, 4532973676, 48095557700, 519247705968, 5690272928520, 63172884082028, 709373555125356, 8046263496489260, 92089662771965492, 1062482514810065752
Offset: 0

Views

Author

Paul D. Hanna, Nov 23 2009

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 20*x^3 + 112*x^4 + 768*x^5 +...
A(x)^(1/2) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 322*x^5 + 2546*x^6 +...+ A006664(n)*x^n +...
G.f. satisfies: A(x*F(x)^2) = F(x)^2 where F(x) = g.f. of A001246:
F(x) = 1 + x + 4*x^2 + 25*x^3 + 196*x^4 + 1764*x^5 + 17424*x^6 +...+ A000108(n)^2*x^n +...
F(x)^2 = 1 + 2*x + 9*x^2 + 58*x^3 + 458*x^4 + 4120*x^5 + 40569*x^6 +...+ A168358(n)*x^n +...

Crossrefs

Cf. A168358, A168344, A001246, A006664, A000108.

Programs

  • PARI
    {a(n)=local(C_2=vector(n+1, m, (binomial(2*m-2, m-1)/m)^2)); polcoeff(x/serreverse(x*Ser(C_2)^2), n)}

Formula

G.f.: A(x) = x/Series_Reversion(x*F(x)^2) where F(x) = g.f. of A001246, which is the squares of Catalan numbers.
G.f.: A(x) = F(x/A(x))^2 where A(x*F(x)^2) = F(x)^2 where F(x) = g.f. of A001246.

A168358 Self-convolution square of A001246, which is the squares of Catalan numbers.

Original entry on oeis.org

1, 2, 9, 58, 458, 4120, 40569, 426842, 4723890, 54402904, 646992474, 7900772120, 98642862232, 1254984808672, 16227116787737, 212790354730842, 2824992774357362, 37915366854924952, 513837166842215970
Offset: 0

Views

Author

Paul D. Hanna, Nov 23 2009

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 58*x^3 + 458*x^4 + 4120*x^5 +...
A(x)^(1/2) = 1 + x + 4*x^2 + 25*x^3 + 196*x^4 + 1764*x^5 + 17424*x^6 +...+ A001246(n)*x^n +...
A(x) satisfies: A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A006664:
G(x) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 322*x^5 + 2546*x^6 +...+ A006664(n)*x^n +...
G(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 112*x^4 + 768*x^5 + 5984*x^6 +...+ A168357(n)*x^n +...

Crossrefs

Cf. A168357, A168344, A001246, A006664, A000108.

Programs

  • Mathematica
    Table[Sum[CatalanNumber[k]^2 * CatalanNumber[n-k]^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 10 2018 *)
  • PARI
    {a(n)=local(C_2=vector(n+1, m, (binomial(2*m-2, m-1)/m)^2)); polcoeff(Ser(C_2)^2, n)}

Formula

G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)^2) where G(x) = g.f. of A006664, which is the number of irreducible systems of meanders.
G.f.: A(x) = G(x*A(x))^2 where A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A006664.
From Vaclav Kotesovec, Mar 10 2018: (Start)
Recurrence: (n+1)^2*(n+2)^3*(4*n^2 - 5*n - 3)*a(n) = 4*(n+1)^2*(48*n^5 - 12*n^4 - 136*n^3 + 15*n^2 + 49*n - 30)*a(n-1) - 32*(96*n^7 - 312*n^6 + 104*n^5 + 580*n^4 - 630*n^3 + 80*n^2 + 91*n - 12)*a(n-2) + 1024*(n-2)^3*(2*n - 3)^2*(4*n^2 + 3*n - 4)*a(n-3).
a(n) ~ (4/Pi - 1) * 2^(4*n + 3) / (Pi*n^3). (End)

A168594 G.f. A(x) satisfies: A(x) = F(x/A(x)) where A(x*F(x)) = F(x) = g.f. of A133053, which is the squares of Motzkin numbers (A001006).

Original entry on oeis.org

1, 1, 3, 6, 20, 70, 302, 1386, 6902, 35862, 194202, 1082642, 6191680, 36141118, 214715244, 1294849186, 7911159522, 48888093910, 305165808290, 1921992409066, 12202404037088, 78031629139246, 502263432618224, 3252160882871210
Offset: 0

Views

Author

Paul D. Hanna, Dec 01 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 20*x^4 + 70*x^5 + 302*x^6 +...
A(x) satisfies: A(x*F(x)) = F(x) = g.f. of A133053:
F(x) = 1 + x + 4*x^2 + 16*x^3 + 81*x^4 + 441*x^5 + 2601*x^6 +...+ A001006(n)^2*x^n +...

Crossrefs

Cf. A001006, A133053, A168344 (variant).

Programs

  • PARI
    {a(n)=if(n==0,1,polcoeff(x/serreverse(x*sum(m=0,n,polcoeff((1/x)*serreverse(x/(1+x+x^2+x^2*O(x^m))), m)^2 *x^m)+x^2*O(x^n)),n))}

Formula

G.f.: A(x) = x/Series_Reversion(x*F(x)) where F(x) = g.f. of A133053.

A248383 Number of n-strand braids of length at most 3 in the dual monoid B_n^{+*}.

Original entry on oeis.org

1, 5, 177, 2856, 147855, 9845829
Offset: 1

Views

Author

N. J. A. Sloane, Oct 15 2014

Keywords

Links

Crossrefs

Cf. A000108, A168344, A248382-A248386.

A248384 Number of n-strand braids of length at most 4 in the dual monoid B_n^{+*}.

Original entry on oeis.org

1, 6, 367, 14122, 1917046, 356470124
Offset: 1

Views

Author

N. J. A. Sloane, Oct 15 2014

Keywords

Links

Crossrefs

Cf. A000108, A168344, A248382-A248386.

A248385 Number of n-strand braids of length at most 5 in the dual monoid B_n^{+*}.

Original entry on oeis.org

1, 7, 749, 68927, 24672817
Offset: 1

Views

Author

N. J. A. Sloane, Oct 15 2014

Keywords

Links

Crossrefs

Cf. A000108, A168344, A248382-A248386.
Showing 1-9 of 9 results.

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

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