A131090 -id:A131090 - cp's OEIS frontend

A135261 a(n) = 3*A131090(n) - A131090(n+1).

-1, 3, -1, 2, -1, 5, 6, 17, 27, 58, 111, 229, 454, 913, 1819, 3642, 7279, 14565, 29126, 58257, 116507, 233018, 466031, 932069, 1864134, 3728273, 7456539, 14913082, 29826159, 59652325, 119304646, 238609297, 477218587, 954437178, 1908874351, 3817748709, 7635497414

Offset: 0

Paul Curtz, Dec 01 2007

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

a:=[-1,2,-1,2];; for n in [5..40] do a[n]:=2*a[n-1] -a[n-3] +2*a[n-4]; od; a; # G. C. Greubel, Nov 21 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)^2*(1-3*x)/((2*x-1)*(1+x)*(x^2-x+1)) )); // G. C. Greubel, Nov 21 2019

seq(coeff(series((1-x)^2*(1-3*x)/((2*x-1)*(1+x)*(x^2-x+1)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Nov 21 2019

LinearRecurrence[{2,0,-1,2}, {-1,3,-1,2}, 40] (* G. C. Greubel, Oct 07 2016 *)

my(x='x+O('x^40)); Vec((1-x)^2*(1-3*x)/((2*x-1)*(1+x)*(x^2-x+1))) \\ G. C. Greubel, Nov 21 2019

def A135261_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)^2*(1-3*x)/((2*x-1)*(1+x)*(x^2-x+1))).list()
A135261_list(40) # G. C. Greubel, Nov 21 2019

A131090(n) - a(n) = A131556(n).
O.g.f.: (1-x)^2*(1-3*x)/((2*x-1)*(1+x)*(x^2-x+1)). - R. J. Mathar, Jul 22 2008
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4). - G. C. Greubel, Oct 07 2016

Edited and extended by R. J. Mathar, Jul 22 2008

A131088 2*A051731 - A054525 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 3, 0, 1, 2, 3, 0, 1, 3, 0, 0, 0, 1, 1, 3, 3, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 2, 2, 0, 3, 0, 0, 0, 1, 2, 0, 3, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 3, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 2, 3, 0, 3, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jun 14 2007

Row sums = A131089.

A131090: (1, 3, 3, 2, 3, 1, 3, 2, 2, 1, ...) in every column interspersed with (k-1) zeros.

			First few rows of the triangle:
  1;
  3, 1;
  3, 0, 1;
  2, 3, 0, 1;
  3, 0, 0, 0, 1;
  1, 3, 3, 0, 0, 1
  3, 0, 0, 0, 0, 0, 1;
  2, 2, 0, 3, 0, 0, 0, 1;
  ...

Cf. A129979 (left border), A131089 (row sums), A051731, A054525.

T(n,k) = 2*!(n%k) - if (!(n % k), moebius(n/k), 0);
row(n) = vector(n, k, T(n,k));
lista(nn) = for (n=1, nn, v = row(n); for (k=1, #v, print1(v[k], ", "))); \\ Michel Marcus, Feb 26 2022

More terms from Michel Marcus, Feb 26 2022

A225887 a(n) = A212205(2*n + 1).

Original entry on oeis.org

1, 4, 18, 86, 426, 2162, 11166, 58438, 309042, 1648154, 8851206, 47813790, 259585002, 1415431266, 7747200558, 42545600310, 234346445154, 1294260644906, 7165245015510, 39754745775886, 221009855334426, 1230909476804594, 6867024985408638, 38369226561522086

Offset: 0

Michael Somos, May 19 2013

From Peter Bala, Apr 23 2017: (Start)
a(n) is also the number of Schröder paths of semilength n (paths from (0, 0) to (2*n, 0), using only single steps northeast or southeast (steps (1, 1) or (1, -1)) or double steps east (steps (2, 0)), that never fall below the x-axis) in which the (2,0)-steps that are on the horizontal axis come in 3 colors (see Oste and Van der Jeugt, Section 7).
Example: a(2) = 18 because from the origin to the point (4,0) we have 3^2 = 9 paths of type HH, 3 paths of type HUD, 3 paths of type UDH as well as the paths UDUD, UUDD, and UHD.
It follows that the sequence may be calculated as the leading diagonal of the lower triangular array (T(n,k))n,k>=0 defined by the relations: T(n,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k-1) for 1 <= k <= n-1 and T(n,n) = 3*T(n-1,n-1) + T(n,n-1). The array begins: [1], [1, 4], [1, 6, 18], [1, 8, 32, 86], [1, 10, 50, 168, 426].  (End)

			1 + 4*x + 18*x^2 + 86*x^3 + 426*x^4 + 2162*x^5 + 11166*x^6 + 58438*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
Taras Goy and Mark Shattuck, Hessenberg-Toeplitz Matrix Determinants with Schröder and Fine Number Entries, Carpathian Math. Publ., Vol. 15 (2023), No. 2, 420-436. See Theorems 1 and 2.
R. Oste and J. Van der Jeugt, Motzkin paths, Motzkin polynomials and recurrence relations, Electronic Journal of Combinatorics 22(2) (2015), #P2.8. Section 7.

Cf. A001003, A006125, A127850, A131090, A151090, A212205, A111966.

a[ n_] := SeriesCoefficient[ 2 / (1 - 5 x + Sqrt[1 - 6 x + x^2]), {x, 0, n}]

a(n):=sum((k+1)*sum(binomial(j,n-k-j)*3^(-n+k+2*j)*2^(n-k-j)*binomial(n+1,j),j,0,n+1),k,0,n)/(n+1); /* Vladimir Kruchinin, Mar 13 2016 */

{a(n) = if( n<0, 0, polcoeff( 2 / (1 -  5*x + sqrt(1 - 6*x + x^2 + x * O(x^n))), n))}

G.f.: (-1 + 5*x + sqrt(1 - 6*x + x^2)) / (2 * (x - 6*x^2)) = 2 / (1 - 5*x + sqrt(1 - 6*x + x^2)).
G.f.: A(x) = 1 / (1 - 5*x + (x - 6*x^2) * A(x)) = 1 + x * A(x) * (5 - A(x) * (1 - 6*x)).
INVERT transform of A001003(n+1). INVERT transform is A134425.
HANKEL transform is A006125. HANKEL transform with 1 prepended is A127850(n+1).
BINOMIAL transform of A151090.
Conjecture: (n+1)*a(n) +3*(-4*n-1)*a(n-1) +(37*n-20)*a(n-2) +6*(-n+2)*a(n-3)=0. - R. J. Mathar, May 23 2014
a(n) = Sum_{k=0..n}((k+1)*Sum_{j=0..n+1}(binomial(j,n-k-j)*3^(-n+k+2*j)*2^(n-k-j)*binomial(n+1,j)))/(n+1). - Vladimir Kruchinin, Mar 13 2016
a(n) ~ (1+sqrt(2))^(2*n+5) / (2^(3/4)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 13 2016
G.f.: 1/(1-3*x -x/(1-x -x/(1-x -x/(1-x - ... )))) (continued fraction) = 1/(1 - 3*x - x*S(x)), where S(x) is the generating function of the large Schröder numbers A001003. - Peter Bala, Apr 23 2017

A131089 a(n) = Sum_{d|n} (2 - mu(d)).

Original entry on oeis.org

1, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 12, 4, 8, 8, 10, 4, 12, 4, 12, 8, 8, 4, 16, 6, 8, 8, 12, 4, 16, 4, 12, 8, 8, 8, 18, 4, 8, 8, 16, 4, 16, 4, 12, 12, 8, 4, 20, 6, 12, 8, 12, 4, 16, 8, 16, 8, 8, 4, 24, 4, 8, 12, 14, 8, 16, 4, 12, 8, 16, 4, 24, 4, 8, 12, 12, 8, 16

Offset: 1

Gary W. Adamson, Jun 14 2007

nonn

Apart from the first term the same as A062011. - Andrew Howroyd, Aug 09 2018

			a(4) = 6 = (2 + 3 + 0 + 1), row 4 of A131088.

Row sums of triangle A131088.

Cf. A062011, A051731, A131090, A228483.

[1] cat [2*NumberOfDivisors(n) : n in [2..100] ]; // Vincenzo Librandi, Aug 10 2018

Join[{1}, Table[2 DivisorSigma[0, n], {n, 100}]] (* Vincenzo Librandi, Aug 10 2018 *)

a(n)={2*numdiv(n) - (n==1)} \\ Andrew Howroyd, Aug 09 2018

a(n) = 2*tau(n) = A062011(n) for n > 1. - Andrew Howroyd, Aug 09 2018

Inverse Moebius transform of A228483.

Name changed and terms a(10) and beyond from Andrew Howroyd, Aug 09 2018

cp's OEIS Frontend

A135261 a(n) = 3*A131090(n) - A131090(n+1).

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A131088 2*A051731 - A054525 as infinite lower triangular matrices.

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A225887 a(n) = A212205(2*n + 1).

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A131089 a(n) = Sum_{d|n} (2 - mu(d)).

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