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.

A114587 Number of peaks at odd levels in all hill-free Dyck paths of semilength n+3 (a hill in a Dyck path is a peak at level 1).

Original entry on oeis.org

1, 4, 17, 68, 269, 1056, 4132, 16144, 63046, 246228, 962019, 3760700, 14710589, 57581696, 225546488, 884059808, 3467476430, 13608852968, 53443415522, 210000136136, 825630208466, 3247733377664, 12781815016232, 50328168273408
Offset: 0

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Author

Emeric Deutsch, Dec 11 2005

Keywords

Examples

			a(1)=4 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UUUUDDDD, UU(UD)(UD)DD, UUDU(UD)DD, UUDUDUDD, UU(UD)DUDD and UUDDUUDD (U=(1,1), D=(1,-1)) we have altogether 4 peaks at odd level (shown between parentheses).

Links

Crossrefs

Cf. A114586, A114590, A114515.

Programs

  • Maple
    G:=(1-2*z-3*z^2-2*z^3-(1-z^2)*sqrt(1-4*z))/2/sqrt(1-4*z)/z^4/(2+z)^2: Gser:=series(G,z=0,32): 1,seq(coeff(Gser,z^n),n=1..26);
  • Mathematica
    CoefficientList[Series[(1-2*x-3*x^2-2*x^3-(1-x^2)*Sqrt[1-4*x])/(2*x^4*(2+x)^2*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

Formula

G.f.: (1 - 2*x - 3*x^2 - 2*x^3 - (1 - x^2)*sqrt(1 - 4*x))/(2*x^4*(2 + x)^2 * sqrt(1 - 4*x)).
a(n) = Sum_{k=0..n+1} k*A114586(n+3,k).
Recurrence: 8*n*(n+4)*a(n) = 2*(15*n^2 + 47*n + 18)*a(n-1) + (9*n^2 + 70*n + 80)*a(n-2) - 2*(n+1)*(2*n+1)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(2*n+6)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012

A114588 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at even levels (0<=k<=n-1; n>=1). A hill in a Dyck path is a peak at level 1.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 3, 1, 1, 3, 6, 6, 2, 1, 7, 17, 18, 11, 3, 1, 17, 48, 58, 40, 18, 4, 1, 43, 134, 186, 150, 76, 27, 5, 1, 110, 380, 590, 540, 325, 130, 38, 6, 1, 286, 1083, 1860, 1915, 1305, 624, 206, 51, 7, 1, 753, 3100, 5844, 6660, 5115, 2772, 1097, 308, 66, 8, 1, 2003
Offset: 1

Views

Author

Emeric Deutsch, Dec 11 2005

Keywords

Comments

Row n has n terms. Row sums are the Fine numbers (A000957). Column 0 yields A114589. Sum(k*T(n,k), k=0..n-1) yields A114590.

Examples

			T(4,3) = 1 because we have U(UD)(UD)(UD)D, where U=(1,1), D=(1,-1) (the peaks at even levels are shown between parentheses).
Triangle begins:
0;
0,   1;
1,   0,  1;
1,   3,  1,  1;
3,   6,  6,  2,  1;
7,  17, 18, 11,  3, 1;
17, 48, 58, 40, 18, 4, 1;

Crossrefs

Cf. A000957, A114586, A114587, A114589, A114590.

Programs

  • Maple
    G:=(1-t*z+2*z^2+3*z-2*t*z^2-sqrt(1-3*z^2-2*z*t+2*z^2*t+z^2*t^2-2*z))/2/z/(2+2*z-t*z-t*z^2+z^2)-1: Gser:=simplify(series(G, z=0, 15)): for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 12 do seq(coeff(t*P[n], t^j), j=1..n) od; # yields sequence in triangular form

Formula

G.f.: G-1, where G = G(t,z) satisfies z(2+2z+z^2-tz-tz^2)G^2+(1+2z)(1+z-tz)G+1+z-tz=0.

A171839 Equal to A171368*A007318.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 3, 2, 1, 0, 0, 6, 8, 3, 1, 0, 0, 15, 22, 15, 4, 1, 0, 0, 36, 68, 52, 24, 5, 1, 0, 0, 91, 198, 191, 100, 35, 6, 1, 0, 0, 232, 586, 651, 425, 170, 48, 7, 1, 0, 0, 603, 1718, 2203, 1656, 820, 266, 63, 8, 1, 0, 0, 1585, 5047, 7285, 6299, 3591, 1435, 392, 80
Offset: 0

Views

Author

Philippe Deléham, Dec 19 2009

Keywords

Comments

Another version of A114586.

Examples

			Triangle begins : 1 ; 0,0 ; 1,0,0 ; 1,1,0,0 ; 3,2,1,0,0 ; 6,8,3,1,0,0 ; ...

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A099323(n+1), A126120(n), A005043(n), A000957(n+1), A117641(n) for x = -2, -1, 0, 1, 2 respectively.
Showing 1-3 of 3 results.

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