A052709 -id:A052709 - cp's OEIS frontend

A240688 Expansion of -(xsqrt(-4x^2-4x+1)-2x^2-3x) / ((x+1)sqrt(-4x^2-4x+1)+ 4x^3+8x^2+3*x-1).

1, 1, 5, 19, 81, 351, 1553, 6959, 31489, 143551, 658305, 3033471, 14034177, 65147135, 303285505, 1415422719, 6620053505, 31021657087, 145613977601, 684537354239, 3222408929281, 15187861143551, 71663163121665

Offset: 0

Vladimir Kruchinin, Apr 10 2014

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

Cf. A052709.

CoefficientList[Series[-(x Sqrt[-4 x^2 - 4 x + 1] - 2 x^2 - 3 x) / ((x + 1) Sqrt[-4 x^2 - 4 x + 1] + 4 x^3 + 8 x^2 + 3 x - 1), {x, 0, 25}], x] (* Vaclav Kotesovec, Apr 12 2014 *)

a(n):=sum((sum(binomial(k,n-k-i)*binomial(k+i-1,i),i,0,n-k))*binomial(n,k),k,0,n);

x='x+O('x^50); Vec(-(x*sqrt(-4*x^2-4*x+1)-2*x^2-3*x) / ((x+1)*sqrt(-4*x^2-4*x+1)+ 4*x^3+8*x^2+3*x-1)) \\ G. C. Greubel, Apr 05 2017

a(n) = Sum_{k=0..n} Sum_{i=0..(n-k)} binomial(k,n-k-i)*binomial(k+i-1,i)*binomial(n,k).
A(x) = x*D'(x)/D(x) where D(x)=(1-sqrt(1-4*x-4*x^2))/(2*(1+x)) is g.f. of A052709.
a(n) ~ 2^(n-1/4) * (1+sqrt(2))^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Apr 12 2014
a(n) = Sum_{i=0..n/2} binomial(n,i)*binomial(2*n-2*i-1,n-2*i). - Vladimir Kruchinin, Mar 10 2015
Conjecture: n*(n-1)*a(n) -(3*n-2)*(n-1)*a(n-1) +2*(-4*n^2+7*n-1)*a(n-2) -4*n*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 14 2016
From Peter Bala, Feb 13 2022: (Start)
The o.g.f. A(x) satisfies the differential equation (8*x^4 + 20*x^3 + 14*x^2 + x - 1)*A(x)' + (8*x^3 + 12*x^2 + 6*x + 3)*A(x) - 2 = 0 with A(0) = 1.
n*a(n) = (n+2)*a(n-1) + (14*n-22)*a(n-2) + (20*n-48)*a(n-3) + (8*n-24)*a(n-4).
Mathar's conjectural third-order recurrence above can be verified using Maple's gfun:-rectodiffeq command.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)
From Peter Bala, Aug 30 2025: (Start)
a(n) = Sum_{0 <= i, j <= n/2} binomial(n, j)*binomial(n+i-1, i)*binomial(n, 2*i+2*j) (verified to satisfy Mathar's third-order recurrence using the MulZeil procedure in Doron Zeilberger's MultiZeilberger Maple package).
Equivalently, a(n) = [x^n] ( (1 + x + x^2 + x^3)/(1 - x^2) )^n. Hence the Gauss congruences hold as stated above. Cf. A288470.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. (End)

A360083 a(n) = Sum_{k=0..n} binomial(5k,n-k) Catalan(k).

Original entry on oeis.org

1, 1, 7, 35, 189, 1092, 6538, 40278, 253730, 1626858, 10582616, 69669273, 463319257, 3107941405, 21004392887, 142882885210, 977562617826, 6722361860888, 46438235933700, 322111000796428, 2242538435656450, 15665017062799230, 109761527468995102

Offset: 0

Seiichi Manyama, Jan 25 2023

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A052709, A073155, A360076, A360082.

Cf. A000108, A360090.

Table[Sum[Binomial[5k,n-k]CatalanNumber[k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jul 13 2025 *)

a(n) = sum(k=0, n, binomial(5*k, n-k)*binomial(2*k, k)/(k+1));

my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+x)^5)))

G.f. A(x) satisfies A(x) = 1/(1 - x * (1+x)^5 * A(x)).

G.f.: 2 / (1 + sqrt( 1 - 4*x*(1+x)^5 )).

A369262 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^3 ).

Original entry on oeis.org

1, 1, 5, 17, 80, 363, 1792, 8969, 46319, 242994, 1296046, 6996163, 38175142, 210162728, 1166020560, 6512854409, 36593709385, 206686641555, 1172856064443, 6683348391034, 38228129813288, 219411037878578, 1263245957786120, 7293833100110787, 42224142505632305

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A052709, A369226.

Cf. A369263, A369264.

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

a(n, s=2, t=3, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+3,k) * binomial(2*n-2*k,n-2*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^2)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A133656 Number of below-diagonal paths from (0,0) to (n,n) using steps (1,0), (0,1) and (2k-1,1), k a positive integer.

Original entry on oeis.org

1, 2, 6, 23, 99, 456, 2199, 10962, 56033, 292094, 1546885, 8299058, 45010492, 246377362, 1359339710, 7551689783, 42206697209, 237156951618, 1338917298708, 7591380528489, 43207023511013, 246773061257046, 1413889039642479, 8124356140582768, 46807462792903984

Offset: 0

Brian Drake, Sep 20 2007

			a(4) = 99 since there are 90 Schroeder paths (A006318) from (0,0) to (4,4) plus DNNEN, DNENN, DENNN, DdNN, DNdN, DNNd, EDNNN, ENDNN and dDNN, where E=(1,0), N=(0,1), D=(3,1) and d=(1,1).

Alois P. Heinz, Table of n, a(n) for n = 0..1276 (first 51 terms from Brian Drake)
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

Cf. A006318, A064641, A052709, A063020, A348474.

Row sums of A201080.

A:=series(RootOf(1+_Z*(x-1)+_Z^2*(x-x^2)+_Z^3*x^2-_Z^4*x^3), x, 21): seq(coeff(A,x,i), i=0..20);

a[n_] := Sum[Binomial[n+k, n] * Sum[Binomial[j, -n - 3k + 2j - 2]* (-1)^(n+k-j+1) * Binomial[n+k+1, j], {j, 0, k+n+1}], {k, 0, n}]/(n+1);
a /@ Range[0, 24] (* Jean-François Alcover, Oct 06 2019, after Vladimir Kruchinin *)

a(n):=sum(binomial(n+k,n)*sum(binomial(j,-n-3*k+2*j-2)*(-1)^(n+k-j+1) *binomial(n+k+1,j),j,0,k+n+1),k,0,n)/(n+1); /* Vladimir Kruchinin, Oct 11 2011 */

G.f. g(x) satisfies: g(x) = 1 + x*g(x)^2+x*g(x)/(1-x^2*g(x)^2).
a(n) = sum(k=0..n, binomial(n+k,n)*sum(j=0..k+n+1, binomial(j,-n-3*k+2*j-2) *(-1)^(n+k-j+1)*binomial(n+k+1,j)))/(n+1). - Vladimir Kruchinin, Oct 11 2011
From Peter Bala, Feb 22 2022: (Start)
G.f. g(x) = (1/x)*series reversion of x*(1 + x)*(1 - x)^2/(1 + x - x^2).
It appears that 1 + x*g'(x)/g(x) = 1 + 2*x + 8*x^2 + 41*x^3 + 220*x^4 + ... is the g.f. of A348474. (End)

A201076 Irregular triangle read by rows: number of {0,2}-shifted Schroeder paths of length n and area k.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 2, 3, 6, 7, 7, 5, 0, 0, 1, 2, 3, 6, 10, 13, 16, 20, 19, 15, 8, 0, 0, 1, 2, 3, 6, 10, 16, 22, 29, 39, 48, 53, 56, 57, 46, 30, 13, 0, 0, 0, 1, 2, 3, 6, 10, 16, 25, 35, 48, 66, 85, 106, 127, 147, 167, 179, 178, 168, 146, 103, 58, 21, 0, 0, 0

Offset: 0

N. J. A. Sloane, Nov 26 2011

			Triangle begins
1
1
1 2 0
1 2 3 3 0
1 2 3 6 7 7 5 0 0
1 2 3 6 10 13 16 20 19 15 8 0 0
...

Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

Row sums give A052709. Rows converge to A101277.

Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201079 {0,2,4,6...}, A201080 {0,1,3,5...}, A201159 {0,1,2}.

gf = Expand /@ FixedPoint[1 + (q x + q^2 x^2) # (Normal@# /. {x :> q^2 x}) + O[x]^8 &, 0];
Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 02 2024 *)

Row 5 corrected, rows 6-7 added by Andrey Zabolotskiy, Jan 02 2024

A260774 Certain directed lattice paths.

Original entry on oeis.org

1, 6, 33, 189, 1107, 6588, 39663, 240894, 1473147, 9058554, 55954395, 346934745, 2157989445, 13459891500, 84152389833, 527224251861, 3309194474451, 20804569738218, 130987600581699, 825796890644895, 5212349717906889, 32935490120006604, 208316726580941037

Offset: 0

N. J. A. Sloane, Jul 30 2015

nonn

See Dziemianczuk (2014) for precise definition.

Alois P. Heinz, Table of n, a(n) for n = 0..1235 (first 101 terms from Lars Blomberg)
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.

Cf. A122868, A064641, A156016.

Cf. A006139, A179191, A052709, A025227.

b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
      `if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
      `if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 21 2021

b[x_, y_] := b[x, y] = If[{x, y} == {0, 0}, 1,
     If[x > 0, Sum[b[x - 1, y + j], {j, -1, 1}], 0] +
     If[y > 0, b[x, y - 1], 0] + If[y < 0, b[x, y + 1], 0]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)

See Dziemianczuk (2014) Equation (33a) with m=1.
From Vaclav Kotesovec, Jul 15 2022: (Start)
Recurrence: (n+1)*(4*n - 3)*a(n) = 6*(4*n^2 - n - 1)*a(n-1) + 3*(n-1)*(4*n + 1)*a(n-2).
a(n) ~ (3 + 2*sqrt(3))^(n+1) / sqrt(6*Pi*n). (End)

More terms from Lars Blomberg, Aug 01 2015

A360272 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3k,k) Catalan(n-3*k).

Original entry on oeis.org

1, 1, 2, 5, 15, 46, 147, 485, 1642, 5669, 19883, 70646, 253755, 919925, 3361546, 12368661, 45786219, 170400470, 637200555, 2392962645, 9021255722, 34128098389, 129519490219, 492966689110, 1881289209003, 7197100511317, 27595769836714, 106032318322517

Offset: 0

Seiichi Manyama, Jan 31 2023

nonn

Cf. A052709, A125305.

Cf. A000108, A360267, A360271, A360274.

A360272 := proc(n)
    add(binomial(n-3*k,k)*A000108(n-3*k),k=0..n/3) ;
end proc:
seq(A360272(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a(n) = sum(k=0, n\4, 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 = 0.2463187933841... is the smallest positive root of the equation1 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

A376486 G.f. satisfies A(x) = 1 / (1 - x^3A(x)^3 (1 + x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 4, 8, 4, 22, 66, 66, 162, 560, 840, 1529, 4985, 9690, 16774, 47349, 107229, 195500, 483000, 1172724, 2311516, 5249556, 12910716, 27299992, 59765400, 144602352, 321554224, 700449496, 1654540452, 3789265198, 8344514618, 19327204006

Offset: 0

Seiichi Manyama, Sep 25 2024

nonn

Index entries for reversions of series

Cf. A005043, A052709, A376487.

A376486 := proc(n)
    add(binomial(4*k,k)*binomial(k,n-3*k)/(3*k+1),k=0..floor(n/3)) ;
end proc:
seq(A376486(n),n=0..70) ;
# R. J. Mathar, Sep 26 2024

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

G.f.: (1/x) * Series_Reversion( x*(1-x^3)/(1+x^4) ).
a(n) = Sum_{k=0..floor(n/3)} binomial(4*k,k) * binomial(k,n-3*k) / (3*k+1).
D-finite with recurrence 243*n*(n-1)*(n+1)*a(n) +81*n*(n-1)*(14*n-31)*a(n-1) +27*(73*n-240)*(n-1)*(n-2)*a(n-2) +36*(-22*n^3-99*n^2+731*n-870)*a(n-3) +48*(-263*n^3+1908*n^2-4449*n+3290)*a(n-4) -128*(n-4)*(230*n^2-1429*n+2205)*a(n-5) -768*(n-5)*(43*n-168)*(n-4)*a(n-6) -9216*(n-5)*(n-6)*(2*n-9)*a(n-7) -4096*(n-5)*(n-6)*(n-7)*a(n-8)=0. - R. J. Mathar, Sep 26 2024

A376487 G.f. satisfies A(x) = 1 / (1 - x^4A(x)^4 (1 + x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 5, 10, 5, 0, 35, 105, 105, 35, 285, 1140, 1710, 1140, 2815, 12650, 25300, 25300, 36401, 145036, 356265, 475020, 588145, 1765666, 4893231, 8115800, 10446245, 23513040, 66875620, 130736800, 187081505, 346058115, 927465240

Offset: 0

Seiichi Manyama, Sep 25 2024

nonn

Index entries for reversions of series

Cf. A005043, A052709, A376486.

a(n) = sum(k=0, n\4, binomial(5*k, k)*binomial(k, n-4*k)/(4*k+1));

G.f.: (1/x) * Series_Reversion( x*(1-x^4)/(1+x^5) ).

a(n) = Sum_{k=0..floor(n/4)} binomial(5*k,k) * binomial(k,n-4*k) / (4*k+1).

A102051 Matrix inverse of triangle A101275 (number of Schröder paths).

Original entry on oeis.org

1, -1, 1, 3, -4, 1, -9, 15, -7, 1, 31, -58, 36, -10, 1, -113, 229, -170, 66, -13, 1, 431, -924, 775, -372, 105, -16, 1, -1697, 3795, -3481, 1939, -691, 153, -19, 1, 6847, -15822, 15542, -9674, 4072, -1154, 210, -22, 1, -28161, 66801, -69276, 47012, -22446, 7606, -1788, 276, -25, 1

Offset: 0

Paul D. Hanna, Dec 27 2004

Row sums are {1,0,0,0...}. Absolute row sums form A006139. Column 0 forms signed A052709. Column 1 forms A102052. Column 2 forms A102053.

			Rows begin:
[1],
[ -1,1],
[3,-4,1],
[ -9,15,-7,1],
[31,-58,36,-10,1],
[ -113,229,-170,66,-13,1],
[431,-924,775,-372,105,-16,1],
[ -1697,3795,-3481,1939,-691,153,-19,1],
[6847,-15822,15542,-9674,4072,-1154,210,-22,1],...
Matrix inverse equals triangle A101275:
[1],
[1,1],
[1,4,1],
[1,13,7,1],
[1,44,34,10,1],...

Cf. A101275, A006139, A052709, A102052, A102053, A039599.

T(n,m):=(-1)^(n-m)*(2*m+1)*(sum((binomial(k,n-k)*binomial(2*k,k-m))/(m+k+1),k,0,n)); /* Vladimir Kruchinin, Apr 18 2015 */

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

G.f.: 2/(1+y+(1-y)*sqrt(1+4*x-4*x^2)).

T(n,m) = (-1)^(n-m)*(2*m+1)*Sum_{k=0..n} C(k,n-k)*C(2*k,k-m)/(m+k+1). - Vladimir Kruchinin, Apr 18 2015

cp's OEIS Frontend

A240688 Expansion of -(x*sqrt(-4*x^2-4*x+1)-2*x^2-3*x) / ((x+1)*sqrt(-4*x^2-4*x+1)+ 4*x^3+8*x^2+3*x-1).

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A360083 a(n) = Sum_{k=0..n} binomial(5*k,n-k) * Catalan(k).

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A369262 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^3 ).

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A133656 Number of below-diagonal paths from (0,0) to (n,n) using steps (1,0), (0,1) and (2k-1,1), k a positive integer.

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A201076 Irregular triangle read by rows: number of {0,2}-shifted Schroeder paths of length n and area k.

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Mathematica

Extensions

A260774 Certain directed lattice paths.

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A360272 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * Catalan(n-3*k).

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A376486 G.f. satisfies A(x) = 1 / (1 - x^3*A(x)^3 * (1 + x)).

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A240688 Expansion of -(xsqrt(-4x^2-4x+1)-2x^2-3x) / ((x+1)sqrt(-4x^2-4x+1)+ 4x^3+8x^2+3*x-1).

A360083 a(n) = Sum_{k=0..n} binomial(5k,n-k) Catalan(k).

A360272 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3k,k) Catalan(n-3*k).

A376486 G.f. satisfies A(x) = 1 / (1 - x^3A(x)^3 (1 + x)).

A376487 G.f. satisfies A(x) = 1 / (1 - x^4A(x)^4 (1 + x)).