A132380 -id:A132380 - cp's OEIS frontend

A023426 a(n) = a(n-1) + Sum_{k=0..n-4} a(k)*a(n-4-k), a(0) = 1. Generalized Catalan Numbers.

1, 1, 1, 1, 2, 4, 7, 11, 18, 32, 59, 107, 191, 343, 627, 1159, 2146, 3972, 7373, 13757, 25781, 48437, 91165, 171945, 325096, 616066, 1169667, 2224355, 4236728, 8082374, 15441719, 29542411, 56590472, 108532322, 208387711, 400551615, 770710831, 1484383399

Offset: 0

Olivier Gérard

Number of lattice paths from (0,0) to (n,0) that stay weakly in the first quadrant and such that each step is either U=(2,1),D=(2,-1), or H=(1,0). E.g. a(5)=4 because we have HHHHH, HUD, UDH and UHD. - Emeric Deutsch, Dec 23 2003

Hankel transform is A132380(n+3). - Paul Barry, May 22 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 10, 19-21.
K. Park and G.S. Cheon, Lattice path counting with a bounded strip restriction

Cf. A000108, A001006, A004148, A006318.

a := n -> hypergeom([(1 - n)/4, (2 - n)/4, (3 - n)/4, -n/4], [2, (1 - n)/2, -n/2], 2^6): seq(simplify(a(n)), n = 0..35); # Peter Luschny, Jul 12 2024

Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-4-k ], {k, 0, n-4} ];
CoefficientList[Series[(1-x-Sqrt[(1-x)^2-4*x^4])/(2*x^4), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

G.f.: [1-z-sqrt((1-z)^2-4z^4)]/[2z^4]. - Emeric Deutsch, Dec 23 2003
From Paul Barry, May 22 2009: (Start)
G.f.: 1/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-... (continued fraction).
G.f.: (1/(1-x))c(x^4/(1-x)^2), c(x) the g.f. of A000108.
a(n) = Sum_{k=0..floor(n/4)} C(n-2k,2k)*A000108(k). (End)
D-finite with recurrence (n+4)*a(n) +(n+4)*a(n-1) -(5*n+8)*a(n-2) +3*n*a(n-3) +4*(2-n)*a(n-4) +12*(3-n)*a(n-5)=0. - R. J. Mathar, Sep 29 2012
a(n) ~ sqrt(3) * 2^(n+3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 01 2014
G.f. A(x) satisfies: A(x) = (1 + x^4 * A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jul 20 2021
a(n) = hypergeom([(1 - n)/4, (2 - n)/4, (3 - n)/4, -n/4], [2, (1 - n)/2, -n/2], 64). - Peter Luschny, Jul 12 2024

Name extended by a formula from the author in Mathematica by Peter Luschny, Jul 13 2024

A107851 Expansion of g.f. x(-1-x-3x^2-x^3+2x^5)/((2x^3+x^2-1)*(x^4+1)).

Original entry on oeis.org

0, 1, 1, 4, 4, 5, 9, 10, 18, 29, 41, 68, 100, 149, 233, 346, 530, 813, 1225, 1876, 2852, 4325, 6601, 10026, 15250, 23229, 35305, 53732, 81764, 124341, 189225, 287866, 437906, 666317, 1013641, 1542132, 2346276, 3569413, 5430537, 8261962, 12569362

Offset: 0

Creighton Dement, May 25 2005

Floretion Algebra Multiplication Program, FAMP Code: 1jesforzapseq[(.5i' + .5j' + .5'ki' + .5'kj')*(.5'i + .5'j + .5'ik' + .5'jk')], 1vesforzap = A000004

Index entries for linear recurrences with constant coefficients, signature (0,1,2,-1,0,1,2).

Cf. A107849, A107850, A107852.

CoefficientList[Series[x(-1-x-3x^2-x^3+2x^5)/((2x^3+x^2-1)(x^4+1)), {x,0,50}],x] (* or *) LinearRecurrence[{0,1,2,-1,0,1,2},{0,1,1,4,4,5,9},51] (* Harvey P. Dale, Jul 19 2011 *)

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 2,1,0,-1,2,1,0]^n*[0;1;1;4;4;5;9])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = A159284(n+1) + A132380(n+7).

a(0)=0, a(1)=1, a(2)=1, a(3)=4, a(4)=4, a(5)=5, a(6)=9, a(n)= a(n-2)+ 2*a(n-3)-a(n-4)+a(n-6)+2*a(n-7). - Harvey P. Dale, Jul 19 2011

A109247 Expansion of (1 - 3x^2 - 3x^3 + x^4)/(1 + x^4).

Original entry on oeis.org

1, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3

Offset: 0

Paul Barry, Jun 23 2005

Row sums of Riordan array (1-x-2x^2,x(1-x)), A109246.

After the initial terms, cyclic with period 8: [0,0,-3,-3,0,0,3,3]. - Antti Karttunen, Aug 12 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Cf. A109246, A132380.

CoefficientList[Series[(1-3x^2-3x^3+x^4)/(1+x^4),{x,0,90}],x] (* or *) Join[{1},LinearRecurrence[{0,0,0,-1},{0,-3,-3,0},90]] (* Harvey P. Dale, Mar 24 2012 *)

Vec((1-3*x^2-3*x^3+x^4)/(1+x^4) + O(x^80)) \\ Jinyuan Wang, Mar 22 2020

(define (A109247 n) (case n ((0) 1) ((1 4) 0) ((2 3) -3) (else (- (A109247 (- n 4)))))) ;; (After Harvey P. Dale's Mar 24 2012 recurrence) - Antti Karttunen, Aug 12 2017

a(0)=1, a(1)=0, a(2)=-3, a(3)=-3, a(4)=0, a(n)=-a(n-4) - Harvey P. Dale, Mar 24 2012

For n > 0, a(n) = -3 * A132380(n). - Antti Karttunen, Aug 12 2017

A144700 Generalized (3,-1) Catalan numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 11, 21, 38, 71, 141, 289, 591, 1195, 2410, 4897, 10051, 20763, 42996, 89139, 185170, 385809, 806349, 1689573, 3547152, 7459715, 15714655, 33161821, 70095642, 148388521, 314562189, 667682057, 1418942341

Offset: 0

Paul Barry, Sep 19 2008

Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(3,-1). Hankel transform has g.f. (1-x^3)/(1+x^4) (A132380 (n+3)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).

Cf. A000108, A014137, A023431, A090344, A132380.

[(&+[Binomial(n-k,3*k)*Catalan(k): k in [0..Floor(n/4)]]): n in [0..40]]; // G. C. Greubel, Jun 15 2022

b[n_, m_]:=a[n, m]=Sum[Binomial[n-k,m*k]*CatalanNumber[k], {k,0,Floor[n/(m+1)]}];
A144700[n_]:= b[n,3]; (* A014137 (m=0), A090344 (m=1), A023431 (m=2) *)
Table[A144700[n], {n, 0, 40}] (* G. C. Greubel, Jun 15 2022 *)

[sum(binomial(n-k,3*k)*catalan_number(k) for k in (0..(n//4))) for n in (0..40)] # G. C. Greubel, Jun 15 2022

G.f.: (1/(1-x)) * c(x^4/(1-x)^3), where c(x) is the g.f. of A000108.
a(n) = Sum_{k=0..floor(n/4)} binomial(n-k, 3*k)*A000108(k).
(n+4)*a(n) = 2*(2*n+5)*a(n-1) - 6*(n+1)*a(n-2) + 2*(2*n-1)*a(n-3) +3*(n-2)*a(n-4) - 2*(2*n-7)*a(n-5). - R. J. Mathar, Nov 16 2011
a(n) = b(n, 3), where b(n, m) = Sum_{k=0..floor(n/(m+1))} binomial(n-k, m*k)*A000108(k). - G. C. Greubel, Jun 15 2022

A123634 Upper half of Hankel determinant number wall for A004148.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 0, 0, 1, 4, 0, 0, -1, 1, 8, 4, -2, -1, -1, 1, 17, 7, 3, -3, -1, -1, 1, 37, 25, 6, -6, -4, 0, 0, 1, 82, 121, -38, -4, -16, 0, 0, 1, 1, 185, 461, 160, -104, -64, -16, 4, 1, 1, 1, 423, 2001, 588, -144, -360, -60, -10, 5, 1, 1, 1, 978, 9225, 360, 1836, -2160, -450, -50, 15, 6, 0, 0, 1

Offset: 0

Michael Somos, Oct 04 2006

			Table is:
n\k  0   1   2   3   4   5   6
--  --  --  --  --  --  --  --
0 |  1
1 |  1   1
2 |  1   1   1
3 |  1   2   0   0
4 |  1   4   0   0  -1
5 |  1   8   4  -2  -1  -1
6 |  1  17   7   3  -3  -1  -1

Cf. A004148, A046978, A132380.

{T(n, k) = my(m); if( k<0 || k>n, 0, matdet( matrix(k, k, i, j, polcoeff( (1 - x + x^2 - sqrt(1 - 2*x - x^2 + x^3*(-2 + x + O(x^(m=i+j+n-k-1))))) / (2*x^2), m))))};

T(n, 0) = 1. T(n, 1) = a(n) if n>0, T(n, 2) = a(n+1)*a(n-1) - a(n)^2 if n>1, T(n, 3) = det([a(n-2), a(n-1), a(n); a(n-1), a(n), a(n+1); a(n), a(n+1), a(n+2)]) if n>2 where a(n) = A004148(n).

T(n, n) = A046978(n+1). T(n+1, n) = A132380(n+2). - Michael Somos, Dec 31 2016

cp's OEIS Frontend

A023426 a(n) = a(n-1) + Sum_{k=0..n-4} a(k)*a(n-4-k), a(0) = 1. Generalized Catalan Numbers.

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A107851 Expansion of g.f. x*(-1-x-3*x^2-x^3+2*x^5)/((2*x^3+x^2-1)*(x^4+1)).

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A109247 Expansion of (1 - 3*x^2 - 3*x^3 + x^4)/(1 + x^4).

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A144700 Generalized (3,-1) Catalan numbers.

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A123634 Upper half of Hankel determinant number wall for A004148.

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A107851 Expansion of g.f. x(-1-x-3x^2-x^3+2x^5)/((2x^3+x^2-1)*(x^4+1)).

A109247 Expansion of (1 - 3x^2 - 3x^3 + x^4)/(1 + x^4).