A157003 -id:A157003 - cp's OEIS frontend

A213221 Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A157004 and g(x) is the g.f. of A157003.

1, 2, 1, 6, 3, 1, 18, 10, 4, 1, 58, 32, 15, 5, 1, 192, 106, 52, 21, 6, 1, 650, 357, 180, 79, 28, 7, 1, 2232, 1222, 624, 288, 114, 36, 8, 1, 7746, 4230, 2178, 1035, 439, 158, 45, 9, 1, 27096, 14770, 7648, 3706, 1642, 643, 212, 55, 10, 1

Offset: 0

Philippe Deléham, Mar 02 2013

			Triangle begins
1
2, 1
6, 3, 1
18, 10, 4, 1
58, 32, 15, 5, 1
192, 106, 52, 21, 6, 1
650, 357, 180, 79, 28, 7, 1
2232, 1222, 624, 288, 114, 36, 8, 1
7746, 4230, 2178, 1035, 439, 158, 45, 9, 1
27096, 14770, 7648, 3706, 1642, 643, 212, 55, 10, 1
95376, 51918, 27000, 13265, 6056, 2508, 911, 277, 66, 11, 1
337404, 183472, 95744, 47532, 22174, 9552, 3708, 1255, 354, 78, 12, 1

Baccherini, D.; Merlini, D.; Sprugnoli, R. Binary words excluding a pattern and proper Riordan arrays. Discrete Math. 307 (2007), no. 9-10, 1021--1037. MR2292531 (2008a:05003). See page 1032. - N. J. A. Sloane, Mar 25 2014

Cf. A157003, A157004 (column k=0), A261058 (column k=1).

Column k has g.f. ((1-sqrt(1-4*x+4*x^3))/(2*(1-x^2)))^k/sqrt(1-4*x+4*x^3).

T(n,0) = 2*T(n,1) - 2*T(n-2,1), T(n+1,k+1) = T(n,k) + T(n+1,k+2) - T(n-1,k+2) for n>=0.

A243753 Number A(n,k) of Dyck paths of semilength n avoiding the consecutive step pattern given by the binary expansion of k, where 1=U=(1,1) and 0=D=(1,-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 2, 1, 4, 1, 1, 0, 0, 0, 1, 1, 2, 4, 1, 9, 1, 1, 0, 0, 0, 1, 1, 2, 4, 9, 1, 21, 1, 1, 0, 0, 0, 1, 1, 1, 4, 9, 21, 1, 51, 1, 1, 0, 0, 0

Offset: 0

Alois P. Heinz, Jun 09 2014

			Square array A(n,k) begins:
  1, 1, 1, 1, 1,   1, 1,   1,   1,    1, ...
  0, 0, 0, 1, 1,   1, 1,   1,   1,    1, ...
  0, 0, 0, 1, 1,   1, 1,   2,   2,    2, ...
  0, 0, 0, 1, 1,   2, 1,   4,   4,    4, ...
  0, 0, 0, 1, 1,   4, 1,   9,   9,    9, ...
  0, 0, 0, 1, 1,   9, 1,  21,  21,   23, ...
  0, 0, 0, 1, 1,  21, 1,  51,  51,   63, ...
  0, 0, 0, 1, 1,  51, 1, 127, 127,  178, ...
  0, 0, 0, 1, 1, 127, 1, 323, 323,  514, ...
  0, 0, 0, 1, 1, 323, 1, 835, 835, 1515, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns give: 0, 1, 2: A000007, 3, 4, 6: A000012, 5: A001006(n-1) for n>0, 7, 8, 14: A001006, 9: A135307, 10: A078481 for n>0, 11, 13: A105633(n-1) for n>0, 12: A082582, 15, 16: A036765, 19, 27: A114465, 20, 24, 26: A157003, 21: A247333, 25: A187256(n-1) for n>0.
Main diagonal gives A243754 or column k=0 of A243752.
Cf. A242450, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.

A:= proc(n, k) option remember; local b, m, r, h;
      if k<2 then return `if`(n=0, 1, 0) fi;
      m:= iquo(k, 2, 'r'); h:= 2^ilog2(k); b:=
      proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
        `if`(t=m and r=1, 0, b(x-1, y+1, irem(2*t+1, h)))+
        `if`(t=m and r=0, 0, b(x-1, y-1, irem(2*t, h)))))
      end; forget(b);
      b(2*n, 0, 0)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_] := A[n, k] = Module[{b, m, r, h}, If[k<2, Return[If[n == 0, 1, 0]]]; {m, r} = QuotientRemainder[k, 2]; h = 2^Floor[Log[2, k]]; b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, If[t == m && r == 1, 0, b[x-1, y+1, Mod[2*t+1, h]]] + If[t == m && r == 0, 0, b[x-1, y-1, Mod[2*t, h]]]]]; b[2*n, 0, 0]]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)

A157004 Transform of central binomial coefficients A000984 whose Hankel transform obeys a Somos-4 recurrence.

Original entry on oeis.org

1, 2, 6, 18, 58, 192, 650, 2232, 7746, 27096, 95376, 337404, 1198546, 4272308, 15273888, 54744268, 196646922, 707747988, 2551624304, 9213416524, 33313656888, 120604436624, 437112790668, 1585877246424, 5759085911154

Offset: 0

Paul Barry, Feb 20 2009

Hankel transform is A157005. Image of A000984 under Riordan array (1,x(1-x^2)).

Diagonal of rational function 1/(1 - x - y + x^3*y^2). - Seiichi Manyama, Mar 23 2023

			G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 58*x^4 + 192*x^5 + 650*x^6 + 2232*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)

Cf. A157003, A360219, A360266.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/Sqrt(1-4*x+4*x^3) )); // G. C. Greubel, Feb 26 2019

CoefficientList[Series[1/Sqrt[1-4*x*(1-x^2)], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

{a(n)=polcoeff(sum(m=0, n, (2*m)!/m!^2 * x^(2*m)*(1-x)^m / (1-2*x+x*O(x^n))^(2*m+1)), n)} \\ Paul D. Hanna, Sep 21 2013

my(x='x+O('x^30)); Vec(1/sqrt(1-4*x+4*x^3)) \\ G. C. Greubel, Feb 26 2019

(1/sqrt(1-4*x+4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

G.f.: 1/sqrt(1 - 4*x*(1 - x^2)).
a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*(1+(-1)^(n-k))*C(k,floor((n-k)/2)) *A000984(k)/2.
G.f.: Sum_{n>=0} (2*n)!/n!^2 * x^(2*n) * (1-x)^n / (1-2*x)^(2*n+1). - Paul D. Hanna, Sep 21 2013
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1) - 2*(2*n-3)*a(n-3). - Vaclav Kotesovec, Feb 13 2014
a(n) ~ (1/r)^n / (sqrt(Pi*n) * sqrt(3-8*r)), where r = 0.2695944364054... is the root of the equation 4*r*(1-r^2)=1. - Vaclav Kotesovec, Feb 13 2014
0 = a(n)*(16*a(n+1) - 32*a(n+3) + 10*a(n+4)) + a(n+1)*(-2*a(n+3)) + a(n+2)*(16*a(n+3) - 6*a(n+4)) + a(n+3)*(-2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Sep 03 2016

A087626 Expansion of 2/(1-2x+sqrt(1-4x+4x^3)).

Original entry on oeis.org

1, 2, 5, 13, 36, 104, 311, 955, 2995, 9553, 30896, 101082, 333946, 1112496, 3732955, 12605029, 42800317, 146046819, 500555447, 1722402303, 5948047169, 20607691517, 71610355540, 249520257106, 871614139396, 3051737703526

Offset: 0

Michael Somos, Sep 16 2003

nonn

			G.f. = 1 + 2*x + 5*x^2 + 13*x^3 + 36*x^4 + 104*x^5 + 311*x^6 + 955*x^7 + ... - _Michael Somos_, Mar 28 2020

Robert Israel, Table of n, a(n) for n = 0..1762
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.

Cf. A006720, A051138, A056010, A157003.

f:= gfun:-rectoproc({(6+4*n)*a(n)+(-6-4*n)*a(n+1)+(-18-4*n)*a(2+n)+(24+5*n)*a(n+3)+(-6-n)*a(n+4), a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 13},a(n),remember):
map(f, [$0..50]); # Robert Israel, Oct 26 2018

CoefficientList[Series[2/(1-2x+Sqrt[1-4x+4x^3]),{x,0,30}],x] (* Harvey P. Dale, Jun 12 2017 *)

{a(n) = polcoeff(2 / (1 - 2*x + sqrt(1 - 4*x + 4*x^3 + x*O(x^n))), n)};

G.f.: 2/(1-2x+sqrt(1-4x+4x^3)).
G.f. A(x) satisfies 0 = x^2*(1-x)*A(x)^2 - (1-2*x)*A(x) + 1.
First backwards difference is A056010.
(6+4*n)*a(n)+(-6-4*n)*a(n+1)+(-18-4*n)*a(2+n)+(24+5*n)*a(n+3)+(-6-n)*a(n+4)=0. - Robert Israel, Oct 26 2018
HANKEL transform is A006720(n+2). HANKEL transform with 0 prepended is -A051138.
INVERT transform of A157003. - Michael Somos, Mar 28 2020

A360271 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3k,k) Catalan(n-3*k).

Original entry on oeis.org

1, 1, 2, 5, 13, 38, 117, 373, 1222, 4085, 13877, 47766, 166229, 583893, 2067414, 7371093, 26440789, 95355990, 345538389, 1257486165, 4593933398, 16841578325, 61938532181, 228454719830, 844882459989, 3132258655573, 11638656376150, 43337083401557

Offset: 0

Seiichi Manyama, Jan 31 2023

nonn

Cf. A157003.

Cf. A000108, A360219, A360272.

A360271 := proc(n)
    add((-1)^k*binomial(n-3*k,k)*A000108(n-3*k),k=0..n/3) ;
end proc:
seq(A360271(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));

my(N=30, x='x+O('x^N)); Vec(2/(1+(sqrt(1-4*x*(1-x^3)))))

G.f.: c(x * (1-x^3)), where c(x) is the g.f. of A000108.
a(n) ~ 2 * sqrt(1-3*r) / (sqrt(Pi) * n^(3/2) * r^n), where r = 0.2541737124933... is the smallest positive root of the equation 1 - 4*r + 4*r^4 = 0. - Vaclav Kotesovec, Feb 01 2023
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-n-1)*a(n-3) +2*(4*n-11)*a(n-4) +4*(-n+5)*a(n-7)=0. - R. J. Mathar, Mar 12 2023

A377264 Consider the recurrence d(k) = (d(k-3)d(k-2) + 1)/(d(k-5)d(k-4)d(k-3)^2d(k-2)^2d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2n+1)).

Original entry on oeis.org

1, 2, 3, 14, 69, 413, 7222, 90211, 2577626, 127385577, 5092018073, 655664812074, 78618294607139, 13276948495989478, 5995083279193033837, 1895278734817024984181, 1542333923096758721461086, 1867485777936169465836858947, 2020248742951852823878208914098, 7078136335206254534330825538868049

Offset: 0

Thomas Scheuerle, Oct 22 2024

nonn

Consider the sequence s(k) with ordinary generating function: 1/(1-d(0)*x/(1-d(1)*x/(1-d(2)*x/(...)))), the Hankel sequence transform of s(k) is A006720 starting with the third term.
This is a special case of a more general theorem: Consider the sequence h(k) with generating function 1/(1-c(0)*x/(1-c(1)*x/(1-c(2)*x/(...)))), if c(2*k+1) = (c(2*k-2)*c(2*k-1) + t)/(c(2*k-4)*c(2*k-3)*c(2*k-2)^2*c(2*k-1)^2*c(2*k)) for all k with c(< 0) = 1, then the Hankel sequence transform of h(k) satisfies a Somos-4 A(1, t) recurrence.
If we would change the start condition into d(0..4) = {1,1,-1,-2,(5/2)}, the expansion of the continued fraction generating function would give us A171416, its Hankel sequence transform is again A006720. There exist infinitely many sequences with the same Hankel sequence transform.

Cf. A006720, A028940.

The Hankel transform is directly related to A006720: A157002, A157003, A160702, A171416, A173992, A173993, A254314.

d(n) = if(n<5, [1,1,1,2,1][n+1], (d(n-3)*d(n-2)+1)/(d(n-5)*d(n-4)*d(n-3)^2*d(n-2)^2*d(n-1)))
a(n) = numerator(d(2*n+1))

a(n) = -numerator(ellmul(ellinit([0, 3, -1, 2, 0]), [-1,0], 2*n+1)[1])

a(n) = A006720(n+1)*A006720(n+3).
denominator(d(2*n+1)) = A006720(n+2)^2.
-a(n)/A006720(n+3)^2 are the x-coordinates of (2*n+1) times [-1,0] on the curve y^2 - y = x^3 + 3*x^2 + 2*x. "Times" means here the multiplication under the elliptic group law.

cp's OEIS Frontend

A213221 Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A157004 and g(x) is the g.f. of A157003.

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A243753 Number A(n,k) of Dyck paths of semilength n avoiding the consecutive step pattern given by the binary expansion of k, where 1=U=(1,1) and 0=D=(1,-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

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A157004 Transform of central binomial coefficients A000984 whose Hankel transform obeys a Somos-4 recurrence.

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Magma

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A087626 Expansion of 2/(1-2x+sqrt(1-4x+4x^3)).

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A360271 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * Catalan(n-3*k).

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A377264 Consider the recurrence d(k) = (d(k-3)*d(k-2) + 1)/(d(k-5)*d(k-4)*d(k-3)^2*d(k-2)^2*d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2*n+1)).

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A360271 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3k,k) Catalan(n-3*k).

A377264 Consider the recurrence d(k) = (d(k-3)d(k-2) + 1)/(d(k-5)d(k-4)d(k-3)^2d(k-2)^2d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2n+1)).