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

Views

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.

A124392 A Fine number related number triangle.

Original entry on oeis.org

1, 2, 1, 7, 2, 1, 24, 8, 2, 1, 86, 28, 9, 2, 1, 314, 103, 32, 10, 2, 1, 1163, 382, 121, 36, 11, 2, 1, 4352, 1432, 456, 140, 40, 12, 2, 1, 16414, 5408, 1732, 536, 160, 44, 13, 2, 1, 62292, 20546, 6608, 2064, 622, 181, 48, 14, 2, 1, 237590, 78436, 25314, 7960, 2429, 714, 203, 52, 15, 2, 1
Offset: 0

Views

Author

Paul Barry, Oct 30 2006

Keywords

Comments

First column is A014300. Second column is A114590. Row sums are A001700. Array is given by (f(x)/(x*sqrt(1-4x)), f(x)) where f(x) is g.f. of Fine numbers A000957.

Examples

			Triangle begins
      1;
      2,    1;
      7,    2,    1;
     24,    8,    2,   1;
     86,   28,    9,   2,   1;
    314,  103,   32,  10,   2,  1;
   1163,  382,  121,  36,  11,  2,  1;
   4352, 1432,  456, 140,  40, 12,  2, 1;
  16414, 5408, 1732, 536, 160, 44, 13, 2, 1;

Links

Crossrefs

Cf. A000957, A001700, A014300, A114590.

Programs

  • GAP
    Flat(List([0..10], n-> List([0..n], k-> Binomial(n-j, k)*Binomial(2*j, n-k) ))); # G. C. Greubel, Dec 25 2019
  • Magma
    [(&+[Binomial(n-j, k)*Binomial(2*j, n-k): j in [0..n-k]]): k in [0..n], n in [0.10]]; // G. C. Greubel, Dec 25 2019
  • Maple
    seq(seq( add(binomial(n-j, k)*binomial(2*j, n-k), j=0..n-k), k=0..n), n=0..10); # G. C. Greubel, Dec 25 2019
  • Mathematica
    Table[Sum[Binomial[n-j, k]*Binomial[2*j, n-k], {j,0,n-k}], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 25 2019 *)
  • PARI
    T(n,k) = sum(j=0, n-k, binomial(n-j, k)*binomial(2*j, n-k)); \\ G. C. Greubel, Dec 25 2019
  • Sage
    [[sum(binomial(n-j, k)*binomial(2*j, n-k) for j in (0..n-k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 25 2019

Formula

Riordan array ( 1/(x*sqrt(1-4*x)) * (1-sqrt(1-4*x))/(3-sqrt(1-4*x)), (1-sqrt(1-4*x))/(3-sqrt(1-4*x)) ).
Number triangle T(n, k) = Sum_{j=0..n-k} C(n-j, k)*C(2*j, n-k).
Showing 1-3 of 3 results.

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