A106566 -id:A106566 - cp's OEIS frontend

A000531 From area of cyclic polygon of 2n + 1 sides.

1, 7, 38, 187, 874, 3958, 17548, 76627, 330818, 1415650, 6015316, 25413342, 106853668, 447472972, 1867450648, 7770342787, 32248174258, 133530264682, 551793690628, 2276098026922, 9373521044908, 38546133661492

Offset: 1

Simon Plouffe

Expected number of matches remaining in Banach's original matchbox problem (counted when empty box is chosen), multiplied by 2^(2*n-1). - Michael Steyer, Apr 13 2001

A conjectured definition: Let 0 < a_1 < a_2 <...

a(n) = total weight of upsteps in all Dyck n-paths (A000108) when each upstep is weighted with its position in the path. For example, the Dyck path UDUUDUDD has upsteps in positions 1,3,4,6 and contributes 1+3+4+6=14 to the weight for Dyck 4-paths. The summand (n-k)*binomial(2*n+1, k) in the Maple formula below is the total weight of upsteps terminating at height n-k, 0<=k<=n-1. - David Callan, Dec 29 2006

Catalan transform of binomial transform of squares. - Philippe Deléham, Oct 31 2008

a(n) is also the number of walks of length 2n in the quarter plane starting and ending at the origin using steps {(1,1),(1,0),(-1,0), (-1,-1)} (which appear in Gessel's conjecture) in which the steps (1,0) and (-1,0) appear exactly once each. - Arvind Ayyer, Mar 02 2009

Equals the Catalan sequence, A000108, convolved with A002457: (1, 6, 30, 140, ...). - Gary W. Adamson, May 14 2009

Total number of occurrences of the pattern 213 (or 132) in all skew-indecomposable (n+2)-permutations avoiding the pattern 123. For example, a(1) = 1, since there is one occurrence of the pattern 213 in the set {213, 132}. - Cheyne Homberger, Mar 13 2013

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I.

T. D. Noe, Table of n, a(n) for n = 1..100
Arvind Ayyer, Towards a human proof of Gessel's conjecture, arXiv:0902.2329 [math.CO], 2009, JIS 12 (2009) 09.4.2
Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017.
F. Bowman, Cyclic pentagons, Math. Gaz. 36, (1952). 244-250. MR0051523.
A. Burstein and S. Elizalde, Total occurrence statistics on restricted permutations, arXiv preprint arXiv:1305.3177 [math.CO], 2013.
C. Homberger, Expected patterns in permutation classes, Electronic Journal of Combinatorics, 19(3) (2012), P43.
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Yasuhiko Kamiyama, The Euler characteristic of the regular spherical polygon spaces, arXiv:1803.05559 [math.GT], 2018.
A. F. Moebius, Über die Gleichungen, mittelst welcher aus den Seiten eines in einen Kreis zu beschreibenden Vielecks der Halbmesser des Kreises und die Fläche des Vielecks gefunden werden, Gesammelte Werke, vol. 1., pp. 407-438.
D. P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly, 102 (1995), 523-530.
Eric Weisstein's World of Mathematics, Cyclic Polygon
Y. Q. Zhao, Introduction to Probability with Applications

Cf. A002457 (Banach's modified matchbox problem), A135404, A002457, A258431.

f := proc(n) sum((n-k)*binomial(2*n+1,k),k=0..n-1); end;

a[n_] := ((2n+1)!/n!^2-4^n)/2; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Dec 07 2011, after Pari *)

PARI
```
a(n)=if(n<1,0,((2*n+1)!/n!^2-4^n)/2)
```

a(n) = ((2n+1)!/((n!)^2)-4^n)/2. - Simon Norton (simon(AT)dpmms.cam.ac.uk), May 14 2001
na(n) = (8n-2)a(n-1) - (16n-8)a(n-2), n>1. - Michael Somos, Apr 18 2003
E.g.f.: 1/2*((1+4*x)*exp(2*x)*BesselI(0, 2*x) + 4*x*exp(2*x)*BesselI(1, 2*x) - exp(4*x)). - Vladeta Jovovic, Sep 22 2003
a(n-1) = 4^n*sum_{k=0..n} binomial(2*k+1, k)*4^(-k) = (2*n+1)*(2*n+3)*C(n) - 2^(2*n+1) (C(n) = Catalan); g.f.: x*c(x)/(1-4*x)^(3/2), c(x): g.f. of Catalan numbers A000108. - Wolfdieter Lang
a(n) = Sum_{k=0..n} A039599(n,k)*k^2, for n>=1. - Philippe Deléham, Jun 10 2007
a(n) = Sum_{k=0..n} A106566(n,k)*A001788(k). - Philippe Deléham, Oct 31 2008
(Conjecture) a(n)=2^(2*n)*sum_{k=1..n} cos(k*Pi/(2*n+1))^2*n. - L. Edson Jeffery, Jan 21 2012

Moebius reference from Michael Somos

A130020 Triangle T(n,k), 0<=k<=n, read by rows given by [1,0,0,0,0,0,0,...] DELTA [0,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938 .

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 14, 0, 1, 5, 14, 28, 42, 42, 0, 1, 6, 20, 48, 90, 132, 132, 0, 1, 7, 27, 75, 165, 297, 429, 429, 0, 1, 8, 35, 110, 275, 572, 1001, 1430, 1430, 0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 0

Offset: 0

Philippe Deléham, Jun 16 2007

Reflected version of A106566.

			Triangle begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  2,   0;
  1, 3,  5,   5,   0;
  1, 4,  9,  14,  14,    0;
  1, 5, 14,  28,  42,   42,    0;
  1, 6, 20,  48,  90,  132,  132,    0;
  1, 7, 27,  75, 165,  297,  429,  429,    0;
  1, 8, 35, 110, 275,  572, 1001, 1430, 1430,    0;
  1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862,  0;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Francesca Aicardi, Catalan triangle and tied arc diagrams, arXiv:2011.14628 [math.CO], 2020.

The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A047072, A059365, A099039, A106566, this sequence.
Diagonals give A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...
Cf. A000108 (Catalan numbers), A106566 (row reversal), A210736.

A130020:= func< n,k | n eq 0 select 1 else (n-k)*Binomial(n+k-1, k)/n >;
[A130020(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 14 2022

T[n_, k_]:= (n-k)Binomial[n+k-1, k]/n; T[0, 0] = 1;
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jun 14 2019 *)

{T(n, k) = if( k<0 || k>=n, n==0 && k==0, binomial(n+k, n) * (n-k)/(n+k))}; /* Michael Somos, Oct 01 2022 */

@CachedFunction
def A130020(n, k):
    if n==k: return add((-1)^j*binomial(n, j) for j in (0..n))
    return add(A130020(n-1, j) for j in (0..k))
for n in (0..10) :
    [A130020(n, k) for k in (0..n)]  # Peter Luschny, Nov 14 2012

T(n, k) = A106566(n, n-k).
Sum_{k=0..n} T(n,k) = A000108(n).
T(n, k) = (n-k)*binomial(n+k-1, k)/n with T(0, 0) = 1. - Jean-François Alcover, Jun 14 2019
Sum_{k=0..floor(n/2)} T(n-k, k) = A210736(n). - G. C. Greubel, Jun 14 2022
G.f.: Sum_{n>=0, k>=0} T(n, k)*x^k*z^n = 1/(1 - z*c(x*z)) where c(z) = g.f. of A000108.

A007854 Expansion of 1/(1 - 3xC(x)), where C(x) = (1 - sqrt(1 - 4x))/(2x) = g.f. for the Catalan numbers A000108.

Original entry on oeis.org

1, 3, 12, 51, 222, 978, 4338, 19323, 86310, 386250, 1730832, 7763550, 34847796, 156503064, 703149438, 3160160811, 14206181382, 63874779714, 287242041528, 1291872728826, 5810776384932, 26138647551564, 117587214581508

Offset: 0

Martin Klazar

Chains in rooted plane trees on n nodes.
The Hankel transform of the aerated sequence with g.f. 1/(1-3x^2c(x^2)) is also 3^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n. - Paul Barry, Jan 20 2007
Binomial transform of A112657. - Philippe Deléham, Nov 25 2007
Row sums of the Riordan matrix (1/sqrt(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))) (A116395). - Emanuele Munarini, Apr 26 2011
Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from C(x) = 1/(1 - x*C(x)) = 1/(1 - 3*x*C(x)) (mod 2). - Peter Bala, Jul 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Martin Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions of arbitrary level, arXiv:math/0405597 [math.CO], 2004.
Index entries for sequences related to rooted trees

Cf. A000108, A000984, A076035, A076036, A067347, A116395, A126694.

CoefficientList[Series[(1+3Sqrt[1-4x])/(4-18x),{x,0,25}],x] (* Emanuele Munarini, Apr 26 2011 *)
nm = 25; t = NestList[Append[Accumulate[#], 3 Total[#]] &, {1}, nm];
Table[t[[n, n]], {n, nm}] (*similar to generating Catalan's triangle A009766*)
(* Li Han, Oct 23 2020 *)

makelist(kron_delta(n,0)+sum(binomial(2*n-k,n-k)*(k*3^k)/(2*n-k),k,1,n),n,0,12); /* Emanuele Munarini, Apr 26 2011 */

a(n) = (9*a(n-1)-3*A000108(n-2))/2 = 3*A049027(n-1) = A067336(n-1)*3/2 = A049027(n-1) + A067336(n-1) = A067347(3, n-1). - Henry Bottomley, Jan 16 2002
a(n) = Sum_{k>=0} A106566(n, k)*3^k. - Philippe Deléham, Aug 11 2005
The Hankel transform of this sequence is A000244 = [1, 3, 9, 27, 81, 243, 729, ...](powers of 3). - Philippe Deléham, Nov 26 2006
a(n) = Sum_{k = 0..n} C(2n,n-k)(2k+1)2^k/(n+k+1). - Paul Barry, Jan 20 2007
a(n) = Sum_{k = 0..n} A039599(n,k)*2^k. - Philippe Deléham, Sep 08 2007
a(n) = Sum_{k = 0..n} A116395(n,k). - Vladimir Kruchinin, Mar 09 2011
From Emanuele Munarini, Apr 26 2011 (Start)
a(n) = Sum_{k = 1..n} C(2*n-k,n-k)*(k*3^k)/(2*n-k), for n>0.
a(n) = (1/4)*(9/2)^n-3*Sum_{k=0..n} C(2*k,k)/(2k-1)*(9/2)^(n-k).
D-finite with recurrence: 2*(n+2)*a(n+2)-(17*n+22)*a(n+1)+18*(2*n+1)*a(n)=0. (End)
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) = upper left term in M^n, M = the infinite square production matrix:
  3, 3, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ... (End)

More terms from Henry Bottomley, Jan 16 2002

A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 2, -2, 1, 0, -5, 5, -3, 1, 0, 14, -14, 9, -4, 1, 0, -42, 42, -28, 14, -5, 1, 0, 132, -132, 90, -48, 20, -6, 1, 0, -429, 429, -297, 165, -75, 27, -7, 1, 0, 1430, -1430, 1001, -572, 275, -110, 35, -8, 1, 0, -4862, 4862, -3432, 2002, -1001, 429, -154, 44, -9, 1, 0, 16796, -16796, 11934, -7072, 3640, -1638

Offset: 0

Paul Barry, Sep 23 2004

Row sums are generalized Catalan numbers A064310. Diagonal sums are 0^n+(-1)^n*A030238(n-2). Inverse is A026729, as number triangle. Columns have g.f. (xc(-x))^k=((sqrt(1+4x)-1)/2)^k.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ... ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938. - Philippe Deléham, May 31 2005

			Rows begin {1}, {0,1}, {0,-1,1}, {0,2,-2,1}, {0,-5,5,-3,1}, ...
Triangle begins
  1;
  0,    1;
  0,   -1,    1;
  0,    2,   -2,   1;
  0,   -5,    5,  -3,    1;
  0,   14,  -14,   9,   -4,   1;
  0,  -42,   42, -28,   14,  -5,  1;
  0,  132, -132,  90,  -48,  20, -6,  1;
  0, -429,  429, -297, 165, -75, 27, -7, 1;
Production matrix is
  0,  1,
  0, -1,  1,
  0,  1, -1,  1,
  0, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1, -1,  1

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (2.10) p. 6.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
Cf. A033184, A000108.
Cf. A106566 (unsigned version), A059365
The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

T[n_, k_]:= If[n == 0 && k == 0, 1, If[n == 0 && k > 0, 0, (-1)^(n + k)*Binomial[2*n - k - 1, n - k]*k/n]];  Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 31 2017 *)

{T(n,k) = if(n == 0 && k == 0, 1, if(n == 0 && k > 0, 0, (-1)^(n + k)*binomial(2*n - k - 1, n - k)*k/n))};
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 31 2017

T(n, k) = (-1)^(n+k)*binomial(2*n-k-1, n-k)*k/n for 0 <= k <= n with n > 0; T(0, 0) = 1; T(0, k) = 0 if k > 0. - Philippe Deléham, May 31 2005

A127632 Expansion of c(x*c(x)), where c(x) is the g.f. for A000108.

Original entry on oeis.org

1, 1, 3, 11, 44, 185, 804, 3579, 16229, 74690, 347984, 1638169, 7780876, 37245028, 179503340, 870374211, 4243141332, 20786340271, 102275718924, 505235129250, 2504876652190, 12459922302900, 62167152967680, 311040862133625

Offset: 0

Paul Barry, Jan 20 2007, Jan 25 2007

Old name was: Expansion of 1/(1 - x*c(x) * c(x*c(x))), where c(x) is the g.f. of A000108.
Hankel transform appears to be A075845.
Catalan transform of Catalan numbers. - Philippe Deléham, Jun 20 2007
Number of functions f:[1,n] -> [1,n] satisfying the condition that, for all i < j, f(j) - (j - i) is not in the interval [1, f(i) - 1]; see the Callan reference. - Joerg Arndt, May 31 2013
This is the number of intervals in the comb posets of Pallo. See the Pallo and Csar et al. references for the definition of these posets. For the proof, see the Aval et al. reference - F. Chapoton, Apr 06 2015
Construct a lower triangular array (T(n,k))n,k>=0 by putting the sequence of Catalan numbers as the first column of the array and completing the remaining columns using the recurrence T(n, k) = T(n, k-1) + T(n-1, k). This sequence will then be the leading diagonal of the array. - Peter Bala, May 13 2017
a(n) is the number of uniquely sorted permutations of length 2n+1 that avoid the patterns 231 and 4132. (A permutation is called uniquely sorted if it has exactly one preimage under West's stack-sorting map.  See the Defant link.) - Colin Defant, Jun 08 2019
a(n) is the number of 132-avoiding permutations of length 3*n whose disjoint cycle decomposition contains only 3-cycles (a,b,c) with a>b>c. See the Archer and Graves reference. - Alexander Burstein, Oct 21 2021

Alois P. Heinz, Table of n, a(n) for n = 0..1000
K. Archer and C. Graves, Pattern-restricted permutations composed of 3-cycles, arXiv:2104.12664 [math.CO], 2021.
J.-C. Aval and F. Chapoton, Poset structures on (m+2)-angulations and polynomial bases of the quotient by Gm-quasisymmetric functions, Séminaire Lotharingien de Combinatoire, vol 77, article B77b.
Peter Bala, A note on the Catalan transform of a sequence
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
David Callan, A combinatorial interpretation of the Catalan transform of the Catalan numbers, arXiv:1111.0996 [math.CO], 2011.
David Callan, Permutations avoiding 4321 and 3241 have an algebraic generating function, arXiv:1306.3193 [math.CO], 2013.
S. Csar, R. Sengupta, and W. Suksompong, On a Subposet of the Tamari Lattice, arXiv preprint arXiv:1108.5690 [math.CO], 2011.
Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.
Tian-Xiao He and Louis W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra and its Applications, Volume 507, 15 October 2016, Pages 77-95.
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
J. M. Pallo, Right-arm rotation distance between binary trees, Inform. Process. Lett., 87(4):173-177, 2003.
Y. Sun and Z. Wang, Consecutive pattern avoidances in non-crossing trees, Graph. Combinat. 26 (2010) 815-832, table 1, {ud}.
S. Yakoubov, Pattern Avoidance in Extensions of Comb-Like Posets, arXiv preprint arXiv:1310.2979 [math.CO], 2013.

Row sums of number triangle A127631.

Cf. A127714, A075845, A000108, A009766, A344056, A344057, A121988.

a:= proc(n) option remember; `if`(n<3, [1, 1, 3][n+1],
      ((8*(4*n-11))*(4*n-5)*(4*n-9)*(2*n-5)*a(n-3)
      -(8*(4*n-5))*(n-1)*(22*n^2-94*n+99)*a(n-2)
      +8*n*(n-1)*(20*n^2-67*n+48)*a(n-1))/
      ((3*(4*n-9))*(n+1)*n*(n-1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 06 2015

a[n_] := Sum[m*(2*n-m-1)!*HypergeometricPFQ[{m/2+1/2, m/2, m-n}, {m, m-2*n+1}, 4]/(n!*(n-m)!), {m, 1, n}]; a[0]=1; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 24 2012, after Vladimir Kruchinin *)
a[n_] := CatalanNumber[n - 1] HypergeometricPFQ[{3/2, 2, 1 - n}, {3, 2 - 2 n}, 4];
a[0] := 1; Table[a[n], {n, 0, 23}] (* Peter Luschny, May 12 2021 *)

a(n):=if n=0 then 1 else sum(m*sum(binomial(2*k-m-1,k-1)*binomial(2*n-k-1,n-1),k,m,n),m,1,n)/n; /* Vladimir Kruchinin, Oct 08 2011 */

{a(n)= if(n<1, n==0, polcoeff( serreverse( x*(1-x)^3*(1-x^3)/(1-x^2)^4 +x*O(x^n) ), n))} /* Michael Somos, May 04 2007 */

{a(n)= local(A); if(n<1, n==0, A= serreverse( x-x^2 +x*O(x^n) ); polcoeff( 1/(1 - subst(A, x, A)), n))} /* Michael Somos, May 04 2007 */

a(n) = A127714(n+1, 2n+1).
G.f. A(x) satisfies: 0 = 1 - A(x) + A(x)^2 * x * c(x) where c(x) is the g.f. of A000108.
G.f.: 2/(1 + sqrt(2 * sqrt(1 - 4*x) - 1)). - Michael Somos, May 04 2007
a(n) = Sum_{k=0..n} A106566(n, k)*A000108(k). - Philippe Deléham, Jun 20 2007
a(n) = (Sum_{m=1..n} (m*Sum_{k=m..n} binomial(2*k-m-1, k-1)*binomial(2*n-k-1, n-1)))/n, a(0)=1. - Vladimir Kruchinin, Oct 08 2011
Conjecture: 3*n*(n-1)*(4*n-9)*(n+1)*a(n) - 8*n*(n-1)*(20*n^2-67*n+48)*a(n-1) + 8*(4*n-5)*(n-1)*(22*n^2-94*n+99)*a(n-2) - 8*(4*n-11)*(4*n-5)*(4*n-9)*(2*n-5)*a(n-3) = 0. - R. J. Mathar, May 04 2018
a(n) ~ 2^(4*n - 1/2) / (sqrt(Pi) * n^(3/2) * 3^(n - 1/2)). - Vaclav Kotesovec, Aug 14 2018
From Alexander Burstein, Nov 21 2019: (Start)
G.f.: A(x) = 1 + x*c(x)^2*m(x*c(x)^2), where m(x) is the g.f. of A001006 and c(x) is the g.f. of A000108.
G.f.: A(x) satisfies: A(-x*A(x)^5) = 1/A(x). (End)
From Peter Luschny, May 12 2021: (Start)
a(n) = Catalan(n - 1) * hypergeom([3/2, 2, 1 - n], [3, 2 - 2*n], 4) for n >= 1.
a(n) = A344056(n) / A344057(n).  (End)
The G.f. satisfies the algebraic equation 0 = F^4*x - F^3 + 2*F^2 - 2*F + 1. - F. Chapoton, Oct 18 2021
D-finite with recurrence 3*n*(n-1)*(n+1)*a(n) -4*n*(7*n-2)*(n-1)*a(n-1) +8*(n-1)*(2*n^2+30*n-65)*a(n-2) +8*(56*n^3-520*n^2+1534*n-1445)*a(n-3) -32*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Aug 01 2022

Better name from David Callan, Jun 03 2013

A076035 G.f.: 1/(1-4xC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

Original entry on oeis.org

1, 4, 20, 104, 548, 2904, 15432, 82128, 437444, 2331128, 12426200, 66250672, 353258536, 1883768176, 10045773072, 53573890464, 285714489348, 1523763466296, 8126565627192, 43341046493424, 231149891614008, 1232790669780816, 6574850950474992, 35065749759115104

Offset: 0

N. J. A. Sloane, Oct 29 2002

nonn

The Hankel transform of this sequence and that of the aerated sequence with g.f. 1/(1-4x^2*c(x^2)) is 4^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n. - Paul Barry, Jan 20 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. B. Ekhad, 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, A000984, A007854, A076036.

CatalanNumber := n -> binomial(2*n,n)/(n+1):
h := (n, m) -> hypergeom([1+m, m-n], [m+n+2], -3):
a := n -> CatalanNumber(n)*(h(n,0) + 6*n/(n+2)*h(n,1)):
seq(simplify(a(n)), n=0..23); # Peter Luschny, Dec 09 2018

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

a(n) = sum{k=0..n, 3^k*C(2n, n-k)(2k+1)/(n+k+1)}. - Paul Barry, Jun 22 2004
a(n) = Sum_{k, 0<=k<=n} A106566(n, k)*4^k. - Philippe Deléham, Sep 01 2005
a(n) = if(n=0,1,sum{k=1..n, C(2n-k-1,n-k)*k*4^k/n}). - Paul Barry, Jan 20 2007
a(n) = Sum{k, 0<=k<=n}A039599(n,k)*3^k. - Philippe Deléham, Sep 08 2007
a(0)=1, a(n)=(16*a(n-1)-4*A000108(n-1))/3. - Philippe Deléham, Nov 27 2007
3*n*a(n) +2*(9-14*n)*a(n-1) +32*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011 [proved by Ekhad & Yang, see link]
a(n) ~ 2^(4*n+1) / 3^(n+1). - Vaclav Kotesovec, Feb 13 2014
Conjecture: a(n) = 4*A076025(n), n>0. - R. J. Mathar, Apr 01 2022

A129442 Expansion of c(x)c(xc(x)) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 6, 21, 80, 322, 1348, 5814, 25674, 115566, 528528, 2449746, 11485068, 54377288, 259663576, 1249249981, 6049846848, 29469261934, 144293491564, 709806846980, 3506278661820, 17385618278700, 86500622296800

Offset: 0

Philippe Deléham, May 28 2007, Jun 20 2007

nonn

The sequence b(n) = [0,1,2,6,21,80,322,1348,...] for n >= 0 is the Catalan transform of Catalan numbers C(n-1), with C(-1)=0; Sum_{k=0..n} A106566(n,k) * A000108(k-1) = b(n).
A121988 is an essentially identical sequence. - R. J. Mathar, Jun 13 2008
Catalan transform of A014137. - R. J. Mathar, Nov 11 2008

			G.f. = 1 + 2*x + 6*x^2 + 21*x^3 + 80*x^4 + 322*x^5 + 1349*x^6 + ... - _Michael Somos_, May 28 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A000108, A014137, A106566, A121988, A236339.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(2*Sqrt(1-4*x)-1))/(2*x) )); // G. C. Greubel, Feb 06 2024

c := proc (x) options operator, arrow; (1/2)*(1-sqrt(1-4*x))/x end proc; G := simplify(c(x)*c(x*c(x))); Gser := series(G, x = 0, 28); seq(coeff(Gser, x, n), n = 0 .. 24) # Emeric Deutsch, Jun 20 2007

a[n_]:= Sum[ Binomial[2n -k-1, n-1]*Binomial[2k-2, k-1], {k, n}]/n;
Array[a, 23] (* Robert G. Wilson v, Jul 18 2007 *)

def A129442_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-sqrt(2*sqrt(1-4*x)-1))/(2*x) ).list()
A129442_list(40) # G. C. Greubel, Feb 06 2024

a(n-1) = (1/n)*Sum_{k=1..n} binomial(2*n-k-1, n-1)*binomial(2*k-2, k-1).
G.f.: (1-sqrt(2*sqrt(1-4*x)-1))/(2*x). - Emeric Deutsch, Jun 20 2007 Corrected by Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007
From Vaclav Kotesovec, Oct 20 2012: (Start)
Recurrence: 3*n*(n+1)*a(n) = 14*n*(2*n-1)*a(n-1) - 4*(4*n-5)*(4*n-3)*a(n-2).
a(n) ~ 2^(4*n+3/2)/(3^(n+1/2)*sqrt(Pi)*n^(3/2)). (End)
0 = +a(n)*(+a(n+1)*(+262144*a(n+2) -275968*a(n+3) +52608*a(n+4)) +a(n+2)*(-50176*a(n+2) +107680*a(n+3) -27930*a(n+4)) +a(n+3)*(-6006*a(n+3) +2574*a(n+4))) +a(n+1)*(+a(n+1)*(-17920*a(n+2) +21952*a(n+3) -4494*a(n+4)) +a(n+2)*(+5152*a(n+2) -15820*a(n+3) +4611*a(n+4)) +a(n+3)*(+1470*a(n+3) -630*a(n+4))) +a(n+2)*(+a(n+2)*(+42*a(n+2) +129*a(n+3) -63*a(n+4)) +a(n+3)*(-63*a(n+3) +27*a(n+4))) for n>=0. - Michael Somos, May 28 2023
From Seiichi Manyama, Jan 10 2023: (Start)
G.f.: (1/x) * Series_Reversion( x * (1-x) * (1-x+x^2) ).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+k,k) * binomial(3*n-k+1,n-2*k). (End)

More terms from Emeric Deutsch, Jun 20 2007

Previous Showing 21-30 of 58 results. Next

This triangle appears in the totally asymmetric exclusion process as Y(alpha=1,beta=1,n,m), written in the Derrida et al. reference as Y_n(m) for alpha=1, beta=1. - Wolfdieter Lang, Jan 13 2006

Number of standard tableaux of shape (n+5,n-4). - Emeric Deutsch, May 30 2004

a(n) is the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly twice. By symmetry, it is also the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x vertically exactly twice. Details can be found in Section 3.3 in Pan and Remmel's link. - Ran Pan, Feb 02 2016

cp's OEIS Frontend

A030237 Catalan's triangle with right border removed (n > 0, 0 <= k < n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A003519 a(n) = 10*C(2n+1, n-4)/(n+6).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A047072 Array A read by diagonals: A(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no step touches the line y=x unless x=0 or x=h.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A000531 From area of cyclic polygon of 2n + 1 sides.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A130020 Triangle T(n,k), 0<=k<=n, read by rows given by [1,0,0,0,0,0,0,...] DELTA [0,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938 .

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A007854 Expansion of 1/(1 - 3*x*C(x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) = g.f. for the Catalan numbers A000108.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A007854 Expansion of 1/(1 - 3xC(x)), where C(x) = (1 - sqrt(1 - 4x))/(2x) = g.f. for the Catalan numbers A000108.

A076035 G.f.: 1/(1-4xC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

A129442 Expansion of c(x)c(xc(x)) where c(x) is the g.f. of A000108.