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.

A167636 Number of peaks at odd level in all Dyck paths of semilength n that have no ascents and no descents of length 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 5, 4, 23, 36, 123, 252, 720, 1664, 4427, 10804, 27971, 69972, 179469, 454300, 1162529, 2961056, 7579620, 19376728, 49659406, 127263208, 326610827, 838550920, 2154985059, 5540935616, 14257159799, 36703613556, 94544579575
Offset: 0

Views

Author

Emeric Deutsch, Nov 08 2009

Keywords

Examples

			a(5)=5 because UUDDUU(UD)DD, UU(UD)DDUUDD, UU(UD)DU(UD)DD, and UUUU(UD)DDDD have 1 + 1 + 2 + 1 = 5 odd-level peaks (shown between parentheses).

Links

Crossrefs

Cf. A167634, A167639.

Programs

  • Maple
    G := (1/2)*z*(1-z^2-2*z^3+z^4-(1+z-z^2)*sqrt((1+z+z^2)*(1-3*z+z^2)))/((1+z-z^2)*sqrt((1+z+z^2)*(1-3*z+z^2))): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);
  • Mathematica
    CoefficientList[Series[1/2*x*(1-x^2-2*x^3+x^4-(1+x-x^2)*Sqrt[(1+x+x^2)*(1-3*x+x^2)])/((1+x-x^2)*Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
  • PARI
    x='x+O('x^50); concat([0,0,0], Vec(1/2*x*(1-x^2-2*x^3+x^4-(1+x-x^2)*sqrt((1+x+x^2)*(1-3*x+x^2)))/((1+x-x^2)*sqrt((1+x+x^2)*(1-3*x+x^2))))) \\ G. C. Greubel, Feb 12 2017

Formula

a(n) = Sum_{k=0..n} k*A167634(n,k).
G.f.: G(z) = z(1 - z^2 - 2z^3 + z^4 - (1 + z - z^2)*sqrt((1 + z + z^2)(1 - 3z + z^2)))/(2(1 + z - z^2)sqrt((1 + z + z^2)(1 - 3z + z^2))).
a(n) ~ sqrt(3/sqrt(5)-1) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+5/2)). - Vaclav Kotesovec, Mar 20 2014
Equivalently, a(n) ~ phi^(2*n - 1) / (4 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
D-finite with recurrence +(n-1)*(38*n-213)*a(n) +38*(n-2)*(n-4)*a(n-1) +4*(-84*n^2+680*n-1185)*a(n-2) -26*(9*n-19)*(n-4)*a(n-3) +(n-5)*(356*n-1879)*a(n-4) +2*(111*n-491)*(n-6)*a(n-5) +2*(95*n-137)*(n-7)*a(n-6) -50*(8*n-23)*(n-8)*a(n-7) +3*(36*n-97)*(n-9)*a(n-8)=0. - R. J. Mathar, Jul 26 2022

A167637 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having k peaks at even level.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 0, 1, 3, 3, 1, 5, 8, 4, 0, 5, 13, 12, 6, 1, 15, 32, 27, 8, 0, 21, 59, 61, 33, 10, 1, 51, 134, 147, 76, 15, 0, 85, 267, 327, 208, 75, 15, 1, 188, 584, 771, 528, 186, 26, 0, 344, 1209, 1734, 1329, 585, 150, 21, 1, 730, 2608, 4008, 3344, 1595, 408, 42, 0
Offset: 0

Views

Author

Emeric Deutsch, Nov 08 2009

Keywords

Comments

Sum of entries in row n is the secondary structure number A004148(n-1) (n>=2).
Row n contains 1 + floor(n/2) entries.
T(n,0) = A167638(n).
Sum_{k>=0} k*T(n,k) = A167639(n).

Examples

			T(6,2)=3 because we have U(UD)DUUU(UD)DDD, UUU(UD)DDDU(UD)D, and UUU(UD)DU(UD)DDD (the even-level peaks are shown between parentheses).
Triangle starts:
  1;
  0;
  0,  1;
  1,  0;
  0,  1,  1;
  2,  2,  0;
  1,  3,  3,  1;
  5,  8,  4,  0;
  5, 13, 12,  6,  1;
  ...

Crossrefs

Cf. A004148, A167634, A167638, A167639.

Programs

  • Maple
    eq := z*(1+z-t*z^2)*G^2-(1+z-z^2)*(1+z-t*z^2)*G+1+z-z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 16 do seq(coeff(P[n], t, j), j = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Formula

G.f.: G=G(t,z) satisfies z(1+z-tz^2)G^2-(1+z-z^2)(1+z-tz^2)G + 1+z-z^2=0.
The trivariate g.f. G=G(t,s,z), where t marks odd-level peaks, s marks even-level peaks, and z marks semilength, satisfies aG^2 - bG + c = 0, where a = z(1+z-sz^2), b=(1+z-tz^2)(1+z-sz^2), c=1+z-tz^2.

A167635 Number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having no peaks at odd level.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 5, 1, 14, 7, 43, 36, 143, 166, 509, 731, 1915, 3158, 7523, 13560, 30537, 58257, 127029, 251266, 538253, 1089666, 2313121, 4754148, 10051130, 20868070, 44065633, 92132176, 194617333, 408971295, 864899013, 1824485600, 3864369141
Offset: 0

Views

Author

Emeric Deutsch, Nov 08 2009

Keywords

Comments

a(n)=A167634(n,0).

Examples

			a(6)=5 because we have UUDDUUDDUUDD, UUDDUUUUDDDD, UUUUDDDDUUDD, UUUUDDUUDDDD, and UUUUUUDDDDDD.

Crossrefs

Cf. A167634, A167638

Programs

  • Maple
    G := ((1+2*z-z^3-sqrt(1-4*z^2-2*z^3+z^6))*1/2)/(z*(1+z-z^2)): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 38);

Formula

G.f.: G = [1 + 2z - z^3 - sqrt(1 - 4z^2 - 2z^3 + z^6)]/[2z(1 + z - z^2)].
D-finite with recurrence (n+1)*a(n) +a(n-1) +(-4*n+5)*a(n-2) +(-2*n+7)*a(n-3) +3*a(n-4) +a(n-5) +(n-6)*a(n-6)=0. - R. J. Mathar, Jul 26 2022
Showing 1-3 of 3 results.

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