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 1-3 of 3 results.

A133365 Number of 3-noncrossing RNA structures, i.e., the number of 3-noncrossing partial matchings over n vertices and without arcs of length 1.

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 105, 321, 1018, 3334, 11216, 38635, 135835, 486337, 1769500, 6531796, 24425758, 92420026, 353444218, 1364933719, 5318450239, 20894505025, 82713826842, 329746065427, 1323179962753, 5341963415921, 21689519880470, 88533441655211
Offset: 1

Views

Author

Emma Y. Jin (emma(AT)cfc.nankai.edu.cn), Oct 26 2007

Keywords

Comments

a(n) is the sum of entries in row n of the triangle in A187253.
a(n) is asymptotically equal to 4!*10.4724*((5+sqrt(21))/2)^n/(n(n-1)(n-2)(n-3)(n-4)).

Examples

			a(4)=5 because we have ABAB, AIAI, AIIA, IAIA, and IIII, where pairs of A's and pairs of B's are assumed to be joined by an arc and the I's are isolated vertices.

Links

Crossrefs

Cf. A000108, A187253.

Programs

  • Maple
    c := proc (n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: T := proc (n, k) if `mod`(n-k, 2) = 0 then sum((-1)^b*binomial(n-b, b)*binomial(n-2*b, k)*(c((1/2)*n-(1/2)*k-b)*c((1/2)*n-(1/2)*k-b+2)-c((1/2)*n-(1/2)*k-b+1)^2), b = 0 .. (1/2)*n-(1/2)*k) else 0 end if end proc: seq(add(T(n, k), k = 0 .. n), n = 1 .. 28);
  • Mathematica
    c = CatalanNumber;
T[n_, k_] := If[EvenQ[m = n-k], Sum[(-1)^b*Binomial[n-b, b] * Binomial[n - 2*b, k] * (c[m/2-b]*c[m/2-b+2] - c[m/2-b+1]^2), {b, 0, m/2}], 0];
a[n_] := Sum[T[n, k], {k, 0, n}];
Array[a, 28] (* Jean-François Alcover, Nov 26 2017, from Maple *)

Formula

a(n) = Sum_{k=0..n} T(n,k), where T(n,k) = Sum((-1)^j*binomial(n-j,j)*binomial(n-2j,k)*[c((n-k)/2-2j)*c((n-k)/2-j+2)-c((n-k)/2-j+1)^2], j=0..(n-k)/2), and c(n)=A000108(n) are the Catalan numbers. [Perhaps this formula is using the convention that c(x) = 0 unless x is a nonnegative integer? - N. J. A. Sloane, Jul 24 2017]

A187255 Number of isolated vertices in all 3-noncrossing RNA structures on n vertices.

Original entry on oeis.org

0, 1, 2, 4, 10, 29, 88, 281, 928, 3170, 11108, 39844, 145700, 542099, 2047390, 7837154, 30359390, 118877686, 470023034, 1874858180, 7538907394, 30538466279, 124544812652, 511112081854, 2109676656860, 8754763717529, 36512355688438, 152987775947396
Offset: 0

Views

Author

Emeric Deutsch, Apr 24 2011

Keywords

Examples

			a(4)=10 because in ABAB, AIAI, AIIA, IAIA, and IIII, where pairs of A's and pairs of B's are assumed to be joined by an arc, we have a total of 0+2+2+2+4=10 isolated vertices (the I's).

Links

Crossrefs

Cf. A187253, A000108.

Programs

  • Maple
    c := proc (n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: T := proc (n, k) if `mod`(n-k, 2) = 0 then sum((-1)^b*binomial(n-b, b)*binomial(n-2*b, k)*(c((1/2)*n-(1/2)*k-b)*c((1/2)*n-(1/2)*k-b+2)-c((1/2)*n-(1/2)*k-b+1)^2), b = 0 .. (1/2)*n-(1/2)*k) else 0 end if end proc: seq(add(k*T(n, k), k = 0 .. n), n = 0 .. 27);

Formula

a(n) = Sum_{k=0..n} k*A187253(n,k).
a(n) = Sum_{k=0..n} k*T(n,k), where T(n,k) = Sum_{j=0..(n-k)/2} (-1)^j*binomial(n-j,j)*binomial(n-2j,k)*(c((n-k)/2 - 2j)*c((n-k)/2-j+2) - c((n-k)/2-j+1)^2), and c(n)=A000108(n) are the Catalan numbers.
a(n) ~ 3 * 21^(3/2) * (5 + sqrt(21))^(n+1) / (Pi * n^4 * 2^(n+4)). - Vaclav Kotesovec, Dec 10 2021

A187254 Number of 3-noncrossing RNA structures over 2n vertices with no isolated vertices.

Original entry on oeis.org

1, 0, 1, 4, 22, 139, 979, 7484, 61018, 523995, 4696277, 43623618, 417729564, 4106089683, 41289287337, 423556384020, 4422308778458, 46904447607369, 504544306691569, 5496706186024364, 60576765646658782, 674624324569952719, 7585425185883023881
Offset: 0

Views

Author

Emeric Deutsch, Apr 24 2011

Keywords

Examples

			a(3) = 4 because we have ABACBC, ABCBAC, ABCACB, and ABCBCA, where identically labeled vertices are assumed to be joined by an arc.

Links

Crossrefs

Cf. A000108, A187253, A358118, A358119.

Programs

  • Maple
    c := n -> binomial(2*n, n)/(n + 1): a := n -> add((-1)^j*binomial(2*n-j, j)*(c(n-j)*c(n-j+2) - c(n-j+1)^2), j = 0..n): seq(a(n), n = 0 .. 22);
  • Mathematica
    Table[Sum[(-1)^j*Binomial[2*n-j,j] * (CatalanNumber[n-j]*CatalanNumber[n-j+2] - CatalanNumber[n-j+1]^2), {j,0,n}], {n,0,25}] (* Vaclav Kotesovec, Dec 10 2021 *)

Formula

a(n) = Sum_{j=0..n} (-1)^j*binomial(2n-j,j)*(c(n-j)*c(n-j+2) - c(n-j+1)^2), where c(i) = A000108(i) are the Catalan numbers.
a(n) = A187253(2*n, 0).
a(n) ~ 27 * (1 + sqrt(3))^(4*n + 2) / (Pi * n^5 * 2^(2*n + 4)). - Vaclav Kotesovec, Dec 10 2021
D-finite with recurrence (n+3)*(n+2)*a(n) + 2*(-7*n^2-2)*a(n-1) + 2*(-2*n+3)*a(n-2) + 2*(7*n^2-42*n+65)*a(n-3) - (n-5)*(n-6)*a(n-4) = 0. - R. J. Mathar, Jul 22 2022
G.f.: (1/4)*(x + 9 - (1 - 14*x + x^2)^(3/2)/x^2*hypergeom([-3/2, 5/2], [2], -16*x/(1 - 14*x + x^2))). - Mark van Hoeij, Nov 10 2022
a(n) = A358119(n) - A358118(n), (see first formula). - Peter Luschny, Nov 11 2022
Maple (depending on the version) gives the third-order recurrence (n - 5)*(2*n - 1)*(n - 4)*a(n - 3) - (n - 1)*(13*n - 24)*(2*n - 3)*a(n - 2) - (n - 1)*(2*n - 1)*(13*n - 2)*a(n - 1) + (n + 3)*(n + 2)*(2*n - 3)*a(n) = 0. - Peter Bala, Nov 11 2022
Showing 1-3 of 3 results.

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