A167638 -id:A167638 - cp's OEIS frontend

A119370 G.f. satisfies A(x) = 1 + xA(x)^2 + x^2(A(x)^2 - A(x)).

1, 1, 2, 6, 19, 64, 225, 816, 3031, 11473, 44096, 171631, 675130, 2679728, 10719237, 43168826, 174885089, 712222799, 2914150406, 11973792218, 49385167369, 204386777160, 848530495383, 3532844222611, 14747626307436, 61712139464939

Offset: 0

Paul D. Hanna, May 16 2006

Equals base sequence of pendular trinomial triangle A119369; iterated convolutions of this sequence with the central terms (A119371) generates all diagonals of A119369. For example: A119372 = A119370 * A119371; A119373 = A119370^2 * A119371.

Diagonal sums of number array A133336. - Philippe Deléham, Nov 09 2009

			A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 225*x^6 + 816*x^7 +...
x*A(x)^2 = x + 2*x^2 + 5*x^3 + 16*x^4 + 54*x^5 + 190*x^6 + 690*x^7 +...
x^2*( A(x)^2 - A(x) ) = 1*x^3 + 3*x^4 + 10*x^5 + 35*x^6 + 126*x^7 +...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 41.

Cf. A104547, A119369, A119371, A119372, A119373, A119374, A119375, A119376.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1+x^2 -Sqrt(1-4*x-2*x^2+x^4))/(2*x*(1+x)) )); // G. C. Greubel, Mar 17 2021

m:= 30;
S:= series( (1+x^2 -sqrt(1-4*x-2*x^2+x^4))/(2*x*(1+x)), x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Mar 17 2021

CoefficientList[Series[((1+x^2)-Sqrt[(1+x^2)^2-4*x*(1+x)])/(2*x*(1+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 11 2013 *)

{a(n)=polcoeff(2/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x*O(x^n))),n)}

def A119370_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1+x^2 -sqrt(1-4*x-2*x^2+x^4))/(2*x*(1+x)) ).list()
A119370_list(30) # G. C. Greubel, Mar 17 2021

G.f.: A(x) = ((1+x^2) - sqrt( (1+x^2)^2 - 4*x*(1+x) ))/(2*x*(1+x)). Equals the inverse binomial transform of A104547.
Recurrence: (n+1)*a(n) = 3*(n-1)*a(n-1) + 6*(n-1)*a(n-2) + 2*(n-2)*a(n-3) - (n-5)*a(n-4) - (n-5)*a(n-5). - Vaclav Kotesovec, Sep 11 2013
a(n) ~ sqrt(-z^2-3*z+1)*(4+2*z-z^3)^(n+1)*(-z^3+z^2+z+3) / (8*sqrt(Pi) * n^(3/2)), where z = 1/(2*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35 + 3*sqrt(129))^(1/3)))) - 1/2*sqrt(8/3-1/3*(280-24*sqrt(129))^(1/3) - 2/3*(35+3*sqrt(129))^(1/3) + 8*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35 + 3*sqrt(129))^(1/3)))) = 0.225270426... is the root of the equation 1-2*z^2+z^4-4*z=0. - Vaclav Kotesovec, Sep 11 2013
G.f.: 1/G(0) where G(k) = 1 - q/(1 - (q + q^2) / G(k+1) ). - Joerg Arndt, Dec 06 2014
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1,k) * binomial(2*n-3*k+1,n-2*k)/(2*n-3*k+1). - Seiichi Manyama, Aug 28 2023
Conjecture: A(x) = 1 + x*exp(Sum_{n >= 1} g(n, x)*x^n/n), where g(n, x) = Sum_{k = 0..n} binomial(n, k)^2*(1 + x)^k. Cf. A105633 and A167638. - Peter Bala, Sep 10 2024

A129509 G.f.: (1+x+x^2-sqrt(1+2x+3x^2-2x^3+x^4))/2.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 2, -4, 3, 5, -20, 29, -1, -94, 221, -191, -327, 1454, -2282, 162, 8002, -19902, 18275, 30505, -143511, 234364, -24437, -841723, 2164873, -2069014, -3325410, 16315410, -27375369, 3714435, 98829168, -260605269, 257026289, 395719442, -2013114895, 3450787313, -572442080, -12414009687, 33423611731, -33865948418, -49805740764, 262037063892

Offset: 0

Paul Barry, Apr 18 2007

sign

Expansion related to the asymptotic mean of the mean square error of a wireless channel.

On page 11 of Tulino and Verdu is equation (1.17) F(x,z) = (sqrt(x(1+sqrt(z))^2+1) - sqrt(x(1-sqrt(z))^2+1))^2. G.f. is F(x,x)/4. - Michael Somos, Mar 18 2014

			G.f. = x^3 - x^4 + 2*x^6 - 4*x^7 + 3*x^8 + 5*x^9 - 20*x^10 + 29*x^11 - x^12 + ...

A. M. Tulino and S. Verdu, Random Matrix Theory and Wireless Communications, Foundations and Trends in Communications and Information Theory, 1 (2004), 1-182.

Cf. A129507.

a[ n_] := SeriesCoefficient[ (1 + x + x^2 - Sqrt[1 + 2*x + 3*x^2 - 2*x^3 + x^4]) / 2, {x, 0, n}]; (* Michael Somos, Mar 18 2014 *)

x='x+O('x^66); Vec( (1+x+x^2-sqrt(1+2*x+3*x^2-2*x^3+x^4))/2 ) \\ Joerg Arndt, Jul 17 2013

G.f.: (sqrt(x(1+sqrt(x))^2+1)-sqrt(x(1-sqrt(x))^2+1))^2/4.
n*a(n) +(2*n-3)*a(n-1) +3*(n-3)*a(n-2) + (9-2*n)*a(n-3) +(n-6)*a(n-4)=0 if n>5. - R. J. Mathar,  Nov 14 2011
Conjecture: g.f.: q^2*(1 - 1/G(0)) where G(k) =  1 + q/(1 + q^2 / G(k+1) ). [Joerg Arndt, Jul 17 2013]
a(n) = A129507(n)/4.
G.f.: 1 + x - (1 + x / (1 + x^2 / (1 + x / (1 + x^2 / ...)))). (continued fraction convergence is three power series terms per iteration) - Michael Somos, Mar 19 2014
G.f.: x * (1 - 1 / (1 - x + x^2 + x / (1 - x + x^2 + x / ...))). (continued fraction convergence is one power series term per iteration) - Michael Somos, Mar 18 2014
G.f.: x^2 * (1 - 1 / (1 + x - x^2 * (1 - 1 / (1 + x - x^2 * (1 - 1 / ...))))). (continued fraction convergence is two power series terms per iteration) - Michael Somos, Mar 30 2014
0 = a(n)*(a(n+1) -5*a(n+2) +12*a(n+3) +11*a(n+4) +7*a(n+5)) + a(n+1)*(a(n+1) -2*a(n+2) -22*a(n+3) -21*a(n+4) -11*a(n+5)) + a(n+2)*(3*a(n+2) +17*a(n+3) +22*a(n+4) +12*a(n+5)) + a(n+3)*(-3*a(n+3) -2*a(n+4) +5*a(n+5)) + a(n+4)*(-a(n+4) +a(n+5)) if n>1. - Michael Somos, Mar 18 2014
Conjecture: g.f. A(x) = x^3 * exp(Sum_{n >= 1} g(n, x)*(-x)^n/n), where g(n, x) = Sum_{k = 0..n} binomial(n, k)^2*x^k. Cf. A167638. - Peter Bala, Sep 10 2024

Prepended a(0)=a(1)=a(2)=0, Joerg Arndt, Jul 17 2013

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

Emeric Deutsch, Nov 08 2009

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).

			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;
  ...

Cf. A004148, A167634, A167638, A167639.

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

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

Emeric Deutsch, Nov 08 2009

nonn

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

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

Cf. A167634, A167638

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);

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

cp's OEIS Frontend

A119370 G.f. satisfies A(x) = 1 + x*A(x)^2 + x^2*(A(x)^2 - A(x)).

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A129509 G.f.: (1+x+x^2-sqrt(1+2x+3x^2-2x^3+x^4))/2.

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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.

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

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A119370 G.f. satisfies A(x) = 1 + xA(x)^2 + x^2(A(x)^2 - A(x)).