A084078 -id:A084078 - cp's OEIS frontend

A084076 Length of list created by n substitutions k -> Range[-1-abs(k), abs(k)+1] starting with {1}.

1, 5, 27, 157, 963, 6141, 40323, 270845, 1852419, 12857341, 90337283, 641286141, 4592533507, 33139654653, 240723001347, 1758796578813, 12916805074947, 95300512382973, 706044251602947, 5250379998560253, 39176121681444867

Offset: 0

Wouter Meeussen, May 11 2003

nonn

2*a(n-1) is the second diagonal of the triangle A115195.

Row sums of A167432. Hankel transform is A167435. - Paul Barry, Nov 03 2009

			{1}
{-2,-1,0,1,2}
{-3,-2,-1,0,1,2,3,-2,-1,0,1,2,-1,0,1,-2,-1,0,1,2,-3,-2,-1,0,1,2,3}

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Third column (m=2) of triangle A115193, called C(1, 2).
Cf. A000108, A115139, A115195, A167432, A167435.
Cf. A084075, A084077, A084078.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-5*x -(1- x)*Sqrt(1-8*x))/(4*x^2*(1+x)) )); // G. C. Greubel, Nov 23 2022

Rest@CoefficientList[InverseSeries[Series[ -((1+5*n+2*n^2-(1+2*n)*Sqrt[1+6*n+n^2] )/(4*n^2)), {n, 0, 28}]], n] or Length/@Flatten/@NestList[ # /. k_Integer :> Range[ -1-Abs[k], Abs[k]+1]&, {1}, 8]
Flatten[{1,RecurrenceTable[{(n+2)*(7*n-5)*a[n] == (7*n-2)*(7*n-1)*a[n-1] + 4*(2*n-1)*(7*n+2)*a[n-2],a[1]==5,a[2]==27},a,{n,20}]}] (* Vaclav Kotesovec, Oct 14 2012 *)

{a(n) = my(L); L = [1]; if(n < 0, 0, for(i = 1, n, L = concat([ vector(3 + 2*abs(k), i, i - abs(k) - 2) | k <- L])); #L)}; /* Michael Somos, Nov 23 2022 */

def A084076_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-5*x -(1-x)*sqrt(1-8*x))/(4*x^2*(1+x)) ).list()
A084076_list(40) # G. C. Greubel, Nov 23 2022

G.f. is the series reversion of -((1 + 5*x + 2*x^2 - (1 + 2*x)*sqrt(1 + 6*x + x^2))/(4*x^2)).
G.f.: 2*((c(2*x))^3)/(1+c(2*x)) with the o.g.f. c(x) of A000108 (Catalan numbers).
a(n) = Sum_{j=1..n+1} A115195(n, j), n >= 0.
G.f.: (-1 + (1-x)*c(2*x))/(x*(1+x)); cf. A115139. - Wolfdieter Lang, Feb 23 2006
D-finite with recurrence: (n+2)*(7*n-5)*a(n) = (7*n-2)*(7*n-1)*a(n-1) + 4*(2*n-1)*(7*n+2)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 7*2^(3n+3)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
D-finite with recurrence (n+2)*a(n) = 2*(4*n+1)*a(n-1) + (n+16)*a(n-2) - 4*(2*n-3)*a(n-3). - R. J. Mathar, Mar 10 2022
a(n) = ( (7*n-1)*(7*n-2)*a(n-1) + 4*(2*n-1)*(7*n+2)*a(n-2) )/((n+2)*(7*n-5)), with a(0) = 1, a(1) = 5. - G. C. Greubel, Nov 23 2022

A084075 Length of list created by n substitutions k -> Range( -abs(k+1), abs(k-1), 2) starting with {1}.

Original entry on oeis.org

1, 2, 5, 12, 33, 86, 249, 680, 2033, 5722, 17485, 50260, 156033, 455534, 1431281, 4228752, 13412193, 40003058, 127840085, 384232156, 1235575201, 3737280582, 12080678505, 36736735672, 119276490193, 364372758986, 1187542872989

Offset: 0

Wouter Meeussen, May 11 2003

nonn

			{1}, {-2,0}, {-1,1,3,-1,1}, {0,2,-2,0,-4,-2,0,2,0,2,-2,0}

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A084076, A084077, A084078.

Cf. A027307, A215067, A034015 (even bisection).

I:=[1,2,5,12]; [n le 4 select I[n] else (6*(35*n^2-55*n-76)*Self(n-1) + (275*n^4-770*n^3-203*n^2+1736*n-912)*Self(n-2) -6*(5*n^2+5*n-28)*Self(n-3) + (n-4)*(n-2)*(25*n^2+5*n-48)*Self(n-4))/(n*(n+2)*(25*n^2-45*n-28)): n in [1..41]]; // G. C. Greubel, Nov 24 2022

Rest@CoefficientList[InverseSeries[Series[ (-1-6n-8n^2+(1+2n)^2 Sqrt[1+4n])/( 2(n+4n^2+4n^3)), {n, 0, 40}]], n]
Length/@Flatten/@NestList[ #/.k_Integer:>Range[-Abs[k+1], Abs[k-1], 2] &, {1}, 8]

# replace iterates lists as described in Example.
def replace(L):
    return [i for k in L for i in range(-abs(k + 1), 1 + abs(k - 1), 2)]
def a(n):
  L = [1]
  for k in range(n): L=replace(L)
  return len(L)
print([a(n) for n in range(12)]) # F. Chapoton, Nov 15 2024

@CachedFunction
def a(n): # a = A084075
    if n < 4: return (1, 2, 5, 12)[n]
    else: return (6*(35*n^2 +15*n -96)*a(n-1) +(275*n^4+330*n^3-863*n^2+120*n+126)*a(n-2) -6*(5*n^2+15*n-18)*a(n-3) +(n-3)*(n-1)*(25*n^2+55*n-18)*a(n-4))/((n+1)*(n+3)*(25*n^2+5*n-48))
[a(n) for n in range(41)] # G. C. Greubel, Nov 24 2022

G.f. is the series reversion of (-1 -6*x -8*x^2 + (1+2*x)^2 * sqrt(1+4*x))/(2*(x +4*x^2 +4*x^3)).
a(2*n) = A027307(n)/2, n >= 1.
a(n) = ( 6*(35*n^2 +15*n -96)*a(n-1) + (275*n^4 +330*n^3 -863*n^2 +120*n +126)*a(n-2) - 6*(5*n^2 +15*n -18)*a(n-3) + (n-3)*(n-1)*(25*n^2 +55*n -18)*a(n-4) )/((n+1)*(n+3)*(25*n^2 +5*n -48)), n >= 4. - G. C. Greubel, Nov 24 2022

A084077 Length of list created by n substitutions k -> Range(-abs(k+1), abs(k-1)) starting with {1}.

Original entry on oeis.org

1, 3, 11, 41, 159, 633, 2575, 10657, 44735, 190017, 815231, 3527681, 15378687, 67478401, 297777407, 1320753665, 5884652543, 26326301697, 118211192831, 532574203905, 2406726828031, 10906541371393

Offset: 0

Wouter Meeussen, May 11 2003

nonn

			{1}, {-2,-1,0}, {-1,0,1,2,3,0,1,2,-1,0,1}

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A084075, A084076, A084078.

I:=[1,3,11]; [n le 3 select I[n] else (3*(7*n^2 -11*n +6)*Self(n-1) + 2*(28*n^2 -51*n +14)*Self(n-2) + 4*(n-2)*(7*n-4)*Self(n-3))/((n+2)*(7*n-11)): n in [1..41]]; // G. C. Greubel, Nov 23 2022

Length/@Flatten/@NestList[ # /. k_Integer:>Range[ -Abs[k+1], Abs[k-1]]&, {1}, 8]
Flatten[{1,RecurrenceTable[{(n+3)*(7*n-4)*a[n] == 3*(7*n^2+3*n+2)*a[n-1] + 2*(28*n^2+5*n-9)*a[n-2] + 4*(n-1)*(7*n+3)*a[n-3],a[1]==3,a[2]==11,a[3]==41},a,{n,20}]}] (* Vaclav Kotesovec, Oct 14 2012 *)

@CachedFunction
def a(n):  # a = A084077
    if (n<3): return (1,3,11)[n]
    else: return (3*(7*n^2 +3*n +2)*a(n-1) + 2*(28*n^2 +5*n -9)*a(n-2) + 4*(n-1)*(7*n+3)*a(n-3))/((n+3)*(7*n-4))
[a(n) for n in range(31)] # G. C. Greubel, Nov 23 2022

invOGF satisfies n - (1+3*n)*a(n) - 2*n*(1+n)*a(n)^2 - 2*n^2*a(n)^3 = 0. [Is it true?]
Recurrence: (n+3)*(7*n-4)*a(n) = 3*(7*n^2+3*n+2)*a(n-1) + 2*(28*n^2+5*n-9)*a(n-2) + 4*(n-1)*(7*n+3)*a(n-3). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(52+34*sqrt(2))*(2+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012

A137842 Number of paths from (0,0) if n is even, or from (2,1) if n is odd, to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).

Original entry on oeis.org

1, 1, 2, 4, 10, 24, 66, 172, 498, 1360, 4066, 11444, 34970, 100520, 312066, 911068, 2862562, 8457504, 26824386, 80006116, 255680170, 768464312, 2471150402, 7474561164, 24161357010, 73473471344, 238552980386, 728745517972

Offset: 0

Paul Barry, Feb 13 2008

Row sums of the inverse of the Riordan array (1/(1+x^2),x(1-x^2)/(1+x^2)).

a(n) is the maximum number of distinct sets that can be obtained as complete parenthesizations of “S_1 union S_2 intersect S_3 union S_4 intersect S_5 union ... S_{n+1}”, where the total of n union and intersection operations alternate, starting with a union, and S_1, S_2, ... , S_{n+1} are sets. - Alexander Burstein, Nov 22 2023

Cf. A084078. [From R. J. Mathar, Feb 28 2009]

G.f.: (1+v^2)/(1-v), where v=2*sqrt(x^2+3)*sin(asin(x(x^2+18)/((x^2+3)^(3/2)))/3)/3-x/3; a(2n)=A027307(n); a(2n+1)=A032349(n+1).

cp's OEIS Frontend

A084076 Length of list created by n substitutions k -> Range[-1-abs(k), abs(k)+1] starting with {1}.

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A084075 Length of list created by n substitutions k -> Range( -abs(k+1), abs(k-1), 2) starting with {1}.

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A084077 Length of list created by n substitutions k -> Range(-abs(k+1), abs(k-1)) starting with {1}.

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Magma

Mathematica

SageMath

Formula

A137842 Number of paths from (0,0) if n is even, or from (2,1) if n is odd, to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).

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Comments

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