A119259 -id:A119259 - cp's OEIS frontend

A064062 Generalized Catalan numbers C(2; n).

1, 1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=2, beta=1 (or alpha=1, beta=2).
a(n) = number of Dyck n-paths (A000108) in which each upstep (U) not at ground level is colored red (R) or blue (B). For example, a(3)=3 counts URDD, UBDD, UDUD (D=downstep). - David Callan, Mar 30 2007
The Hankel transform of this sequence is A002416. - Philippe Deléham, Nov 19 2007
The sequence a(n)/2^n, with g.f. 1/(1-xc(x)/2), has Hankel transform 1/2^n. - Paul Barry, Apr 14 2008
The REVERT transform of the odd numbers [1,3,5,7,9,...] is [1, -3, 13, -67, 381, -2307, 14589, -95235, 636925, ...] - N. J. A. Sloane, May 26 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, example section 3.
Jean-Luc Baril, Sergey Kirgizov, and Mehdi Naima, A lattice on Dyck paths close to the Tamari lattice, arXiv:2309.00426 [math.CO], 2023.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012
J. Bloom and S. Elizalde, Pattern avoidance in matchings and partitions, arXiv:1211.3442 [math.CO] (2012) Theorem 6.1.
N. Bonichon, C. Gavoille and N. Hanusse, Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation, In Proceedings of WG'03, volume 2880 of LNCS, pp. 81-92, 2003.
Alexander Burstein, Sergi Elizalde and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv:math/0610234 [math.CO], 2006.
Xiang-Ke Chang, X.-B. Hu, H. Lei and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
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).
Simone Fioravanti, Steve Hanneke, Shay Moran, Hilla Schefler, and Iska Tsubari, Ramsey Theorems for Trees and a General 'Private Learning Implies Online Learning' Theorem, arXiv:2407.07765 [cs.LG], 2024. See p. 44.
Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
N. J. A. Sloane, Transforms
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011-2012.

Generalized Catalan numbers C(m; n): A000012 (m = 0), A000108 (m = 1), A064063 (m = 3) and A064087 - A064093 (m = 4 thru 10); A064310 (m = -1), A064311 (m = -2) and A064325 - A064333 (m = -3 thru -11).

Cf. A064334, A119529.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (3 - Sqrt(1-8*x))/(2*(1+x)) )); // G. C. Greubel, Sep 27 2024

1, seq(simplify(hypergeom([1-n,n],[-n],2)), n=1..100); # Robert Israel, Nov 30 2014

a[0]=1; a[1]=1; a[n_]/;n>=2 := a[n] = a[n-1] + Sum[(a[k] + a[k-1])a[n-k],{k,n-1}]; Table[a[n],{n,0,10}] (* David Callan, Aug 27 2009 *)
a[n_] := 2*Sum[ (-1)^j*2^(n-j-1)*Binomial[2*(n-j-1), n-j-1]/(n-j), {j, 0, n-1}] + (-1)^n; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 03 2013 *)

{a(n)=polcoeff((3-sqrt(1-8*x+x*O(x^n)))/(2+2*x),n)}

{a(n)=local(A=1+x); for(i=1, n, A=1+A^4*intformal(1/(A^2+x*O(x^n)))); polcoeff(A, n)} \\ Paul D. Hanna, Dec 24 2013
for(n=0, 25, print1(a(n), ", "))

{a(n)=polcoeff(1/(1 - serreverse(x-2*x^2 +x^2*O(x^n))),n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Nov 30 2014

def a(n):
    if n==0: return 1
    return hypergeometric([1-n, n], [-n], 2).simplify()
[a(n) for n in range(22)] # Peter Luschny, Dec 01 2014

G.f.: (1 + 2*x*C(2*x)) / (1+x) = 1/(1 - x*C(2*x)) with C(x) g.f. of Catalan numbers A000108.
a(n) = A062992(n-1) = Sum_{m = 0..n-1} (n-m)*binomial(n-1+m, m)*(2^m)/n, n >= 1, a(0) = 1.
a(n) = Sum_{k = 0..n} A059365(n, k)*2^(n-k). - Philippe Deléham, Jan 19 2004
G.f.: 1/(1-x/(1-2x/(1-2x/(1-2x/(1-.... = 1/(1-x-2x^2/(1-4x-4x^2/(1-4x-4x^2/(1-.... (continued fractions). - Paul Barry, Jan 30 2009
a(n) = (32/Pi)*Integral_{x = 0..1} (8*x)^(n-1)*sqrt(x*(1-x)) / (8*x+1). - Groux Roland, Dec 12 2010
a(n+2) = 8^(n+2)*( c(n+2)-c(1)*c(n+1) - Sum_{i=0..n-1} 8^(-i-2)*c(n-i)*a(i+2) ) with c(n) = Catalan(n+2)/2^(2*n+1). - Groux Roland, Dec 12 2010
a(n) = the upper left term in M^n, M = the production matrix:
  1, 1
  2, 2, 1
  4, 4, 2, 1
  8, 8, 4, 2, 1
  ... - Gary W. Adamson, Jul 08 2011
D-finite with recurrence: n*a(n) + (12-7n)*a(n-1) + 4*(3-2n)*a(n-2) = 0. - R. J. Mathar, Nov 16 2011 (This follows easily from the generating function. - Robert Israel, Nov 30 2014)
G.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^2 dx. - Paul D. Hanna, Dec 24 2013
G.f. satisfies: Integral 1/A(x)^2 dx = x - x^2*G(x), where G(x) is the o.g.f. of A000257, the number of rooted bicubic maps. - Paul D. Hanna, Dec 24 2013
G.f. A(x) satisfies: A(x - 2*x^2) = 1/(1-x). - Paul D. Hanna, Nov 30 2014
a(n) = hypergeometric([1-n, n], [-n], 2) for n > 0. - Peter Luschny, Nov 30 2014
G.f.: (3 - sqrt(1-8*x))/(2*(x+1)). - Robert Israel, Nov 30 2014
a(n) ~ 2^(3*n+1) / (9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 22 2014
O.g.f. A(x) =  1 + series reversion of (x*(1 - x)/(1 + x)^2). Logarithmically differentiating (A(x) - 1)/x gives 3 + 17*x + 111*x^2 + ..., essentially a g.f for A119259. - Peter Bala, Oct 01 2015
From Peter Bala, Jan 06 2022: (Start)
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + ... is a g.f. for A022558.
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)

A014300 Number of nodes of odd outdegree in all ordered rooted (planar) trees with n edges.

Original entry on oeis.org

1, 2, 7, 24, 86, 314, 1163, 4352, 16414, 62292, 237590, 909960, 3497248, 13480826, 52097267, 201780224, 783051638, 3044061116, 11851853042, 46208337584, 180383564228, 704961896036, 2757926215742, 10799653176704, 42326626862636, 166021623024584, 651683311373788

Offset: 1

Emeric Deutsch

Also total number of blocks of odd size in all Catalan(n) possible noncrossing partitions of [n].
Convolution of the sequence of central binomial coefficients 1,2,6,20,70,... (A000984) and of the sequence of Fine numbers 1,0,1,2,6,18,... (A000957).
Row sums of A119307. - Paul Barry, May 13 2006
Hankel transform is A079935. - Paul Barry, Jul 17 2009
Also for n>=1 the number of unimodal functions f:[n]->[n] with f(i)<>f(i+1). a(3) = 7: [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,3,1], [2,3,2], [3,2,1]. - Alois P. Heinz, May 23 2013
Also, number of sets of n rational numbers on [0,1) such that if x belongs to the set, the fractional part of 2x also belongs to it. - Jianing Song and Andrew Howroyd, May 18 2018
Let A(i, j) denote the infinite array such that the i-th row of this array is the sequence obtained by applying the partial sum operator i times to the function ((-1)^(n + 1) + 1)/2 for n > 0. Then A(n, n) equals a(n) for all n > 0. - John M. Campbell, Jan 20 2019
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p >= 3 and positive integers n and k. - Peter Bala, Jan 07 2022

Alois P. Heinz, Table of n, a(n) for n = 1..500
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Hacène Belbachir and Abdelghani Mehdaoui, Diagonal sums in Pascal pyramid (1, 2, r), Les Annales RECITS (2019) Vol. 6, 45-52.
N. Dershowitz and S. Zaks, Ordered trees and non-crossing partitions, Discrete Math., 62 (1986), 215-218.
Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
Index entries for sequences related to rooted trees

Cf. A059481, A000957, A000984, A072547, A119259, A119307, A079935, A333564.

[(&+[(-1)^(n-k)*Binomial(n+k-1, k-1): k in [0..n]]): n in [1..30]]; // G. C. Greubel, Feb 19 2019

a:= proc(n) a(n):= `if`(n<3, n, ((12-40*n+21*n^2) *a(n-1)+
       2*(3*n-1)*(2*n-3) *a(n-2))/ (2*(3*n-4)*n))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 30 2012

Rest[CoefficientList[Series[2x/(1-4x+(1+2x)Sqrt[1-4x]),{x,0,40}],x]]  (* Harvey P. Dale, Apr 25 2011 *)
a[n_] := Sum[Binomial[2k, n-1], {k, 0, n-1}]; Array[a, 30] (* Jean-François Alcover, Dec 25 2015, after Paul Barry *)

a(n) = n--; sum(k=0, n, binomial(2*k,n)); \\ Michel Marcus, May 18 2018

from itertools import count, islice
def A014300_gen(): # generator of terms
    yield from (1,2)
    a, c = 1, 1
    for n in count(1):
        yield (a:=(3*n+5)*(c:=c*((n<<2)+2)//(n+2))-a>>1)
A014300_list = list(islice(A014300_gen(),20)) # Chai Wah Wu, Apr 26 2023

[sum((-1)^(n-k)*binomial(n+k-1, k-1) for k in (0..n)) for n in (1..30)] # G. C. Greubel, Feb 19 2019

a(n) = (binomial(2*n, n) + A000957(n))/3; [simplified by Alexander Burstein, Nov 24 2023]
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1, k-1). - Vladeta Jovovic, Aug 28 2002
G.f.: 2*z/(1-4*z+(1+2*z)*sqrt(1-4*z)).
a(n) = Sum_{j=0..floor((n-1)/2)} binomial(2*n-2*j-2, n-1).
2*a(n) + a(n-1) = (3*n-1)*Catalan(n-1). - Vladeta Jovovic, Dec 03 2004
a(n) = (-1)^n*Sum_{i=0..n} Sum_{j=n..2*n} (-1)^(i+j)*binomial(j, i). - Benoit Cloitre, Jun 18 2005
a(n) = Sum_{k=0..n} C(2*k,n) [offset 0]. - Paul Barry, May 13 2006
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n+k-1,k-1). - Paul Barry, Jul 18 2006
From Paul Barry, Jul 17 2009: (Start)
a(n) = Sum_{k=0..n} C(2*n-k,n-k)*(1+(-1)^k)/2.
a(n) = Sum_{k=0..n} C(n+k,k)*(1+(-1)^(n-k))/2. (End)
a(n) is the coefficient of x^(n+1)*y^(n+1) in 1/(1- x^2*y/((1-2*x)*(1-y))). - Ira M. Gessel, Oct 30 2012
a(n) = -binomial(2*n,n-1)*hyper2F1([1,2*n+1],[n+2], 2). - Peter Luschny, Jul 25 2014
a(n) = [x^n] x/((1 - x^2)*(1 - x)^n). - Ilya Gutkovskiy, Oct 25 2017
a(n) ~ 4^n / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 25 2017
D-finite with recurrence: 2*n*a(n) +(-3*n-4)*a(n-1) +2*(-9*n+19)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Feb 20 2020
a(n) = A333564(n)/2^n. - Peter Bala, Apr 09 2020
a(n) = (1/2)*(binomial(2*n,n) - A072547(n)). - Peter Bala, Mar 28 2023

A119258 Triangle read by rows: T(n,0) = T(n,n) = 1 and for 0

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 7, 1, 1, 7, 17, 15, 1, 1, 9, 31, 49, 31, 1, 1, 11, 49, 111, 129, 63, 1, 1, 13, 71, 209, 351, 321, 127, 1, 1, 15, 97, 351, 769, 1023, 769, 255, 1, 1, 17, 127, 545, 1471, 2561, 2815, 1793, 511, 1, 1, 19, 161, 799, 2561, 5503, 7937, 7423, 4097, 1023, 1

Offset: 0

Reinhard Zumkeller, May 11 2006

From Richard M. Green, Jul 26 2011: (Start)
T(n,n-k) is the (k-1)-st Betti number of the subcomplex of the n-dimensional half cube obtained by deleting the interiors of all half-cube shaped faces of dimension at least k.
T(n,n-k) is the (k-2)-nd Betti number of the complement of the k-equal real hyperplane arrangement in R^n.
T(n,n-k) gives a lower bound for the complexity of the problem of determining, given n real numbers, whether some k of them are equal.
T(n,n-k) is the number of nodes used by the Kronrod-Patterson-Smolyak cubature formula in numerical analysis. (End)

			Triangle begins as:
  1;
  1, 1;
  1, 3,  1;
  1, 5,  7,  1;
  1, 7, 17, 15,  1;
  1, 9, 31, 49, 31, 1;

Reinhard Zumkeller, Rows n=0..120 of triangle, flattened
A. Björner and V. Welker, The homology of "k-equal" manifolds and related partition lattices, Adv. Math., 110 (1995), 277-313.
Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO], 2023.
R. M. Green, Homology representations arising from the half cube, Adv. Math., 222 (2009), 216-239.
R. M. Green, Homology representations arising from the half cube, II, J. Combin. Theory Ser. A, 117 (2010), 1037-1048.
R. M. Green, Homology representations arising from the half cube, II, arXiv:0812.1208 [math.RT], 2008.
R. M. Green and Jacob T. Harper, Morse matchings on polytopes, arXiv preprint arXiv:1107.4993 [math.GT], 2011.
OEIS Wiki, Sequence of the Day for November 3.
K. Petras, On the Smolyak cubature error for analytic functions, Adv. Comput. Math., 12 (2000), 71-93.
M. Shattuck and T. Waldhauser, Proofs of some binomial identities using the method of last squares, Fib. Q., 48 (2010), 290-297.
M. Shattuck and T. Waldhauser, Proofs of some binomial identities using the method of last squares, arXiv:1107.1063 [math.CO], 2010.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A119259, A007318, A007051, A005408, A056220, A027608, A055580, A000337, A000225, A112857 (row reverse).

A145661, A119258 and A118801 are all essentially the same (see the Shattuck and Waldhauser paper). - Tamas Waldhauser, Jul 25 2011

a119258 n k = a119258_tabl !! n !! k
a119258_row n = a119258_tabl !! n
a119258_list = concat a119258_tabl
a119258_tabl = iterate (\row -> zipWith (+)
   ([0] ++ init row ++ [0]) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Nov 15 2011

function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return 2*T(n-1,k-1) + T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 18 2019

# Case m = 2 of the more general:
A119258 := (n,k,m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k, k+1) *hypergeom([1,n+1],[k+2],m)/(k+1)!;
seq(seq(round(evalf(A119258(n,k,2))),k=0..n), n=0..10); # Peter Luschny, Jul 25 2014

T[n_, k_] := Binomial[n, k] Hypergeometric2F1[-k, n-k, n-k+1, -1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 10 2017 *)

T(n,k) = if(k==0 || k==n, 1, 2*T(n-1, k-1) + T(n-1,k) ); \\ G. C. Greubel, Nov 18 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    else: return 2*T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 18 2019

T(2*n,n-1) = T(2*n-1,n) for n > 0;
central terms give A119259; row sums give A007051;
T(n,0) = T(n,n) = 1;
T(n,1) = A005408(n-1) for n > 0;
T(n,2) = A056220(n-1) for n > 1;
T(n,n-4) = A027608(n-4) for n > 3;
T(n,n-3) = A055580(n-3) for n > 2;
T(n,n-2) = A000337(n-1) for n > 1;
T(n,n-1) = A000225(n) for n > 0.
T(n,k) = [k<=n]*(-1)^k*Sum_{i=0..k} (-1)^i*C(k-n,k-i)*C(n,i). - Paul Barry, Sep 28 2007
From Richard M. Green, Jul 26 2011: (Start)
T(n,k) = [k<=n] Sum_{i=n-k..n} (-1)^(n-k-i)*2^(n-i)*C(n,i).
T(n,k) = [k<=n] Sum_{i=n-k..n} C(n,i)*C(i-1,n-k-1).
G.f. for T(n,n-k): x^k/(((1-2x)^k)*(1-x)). (End)
T(n,k) = R(n,k,2) where R(n, k, m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k,k+1)* hyper2F1([1,n+1], [k+2], m)/(k+1)!. - Peter Luschny, Jul 25 2014
From Peter Bala, Mar 05 2018: (Start)
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 + 2*x)^n/(1 + x) about 0. For example, for n = 4 we have (1 + 2*x)^4/(1 + x) = 1 + 7*x + 17*x^2 + 15*x^3 + x^4 + O(x^5).
Row reverse of A112857. (End)

A062992 Row sums of unsigned triangle A062991.

Original entry on oeis.org

1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173

Offset: 0

Wolfdieter Lang, Jul 12 2001

a(n) = N(2; n,x=-1), with the polynomials N(2; n,x) defined in A062991.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
L. Guo and W. Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337.

Cf. A112707 (c(n, -m) triangle). Here m=2 is used. Row sums of A234950.

Cf. A000108, A062991, A064062, A119259.

a062992 = sum . a234950_row  -- Reinhard Zumkeller, Jan 12 2014

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1-2*x-Sqrt(1-8*x))/(2*x+2*x^2) )); // G. C. Greubel, Sep 27 2024

Table[2*Sum[(-1)^j*Binomial[2*n-2*j,n-j]/(n-j+1)*2^(n-j), {j,0,n}]-(-1)^n,{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)

a(n)=polcoeff((1-2*x-sqrt(1-8*x+x^2*O(x^n)))/(2*x+2*x^2),n)

a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x)^2+O(x^(n+2))),n+1)) \\ Ralf Stephan

def a(n): return hypergeometric([-n, n+1], [-n-1], 2)
[a(n).hypergeometric_simplify() for n in range(21)] # Peter Luschny, Nov 30 2014

a(n) = (-1)^(n+1) + 2*Sum_{j = 0..n} (-1)^j*C(n-j)*2^(n-j) with C(n) := A000108(n) (Catalan).
G.f.: A(x) = (2*c(2*x) - 1)/(1 + x) with c(x) the g.f. of A000108.
a(n) = (1/(n+1)) * Sum_{k = 0..n} binomial(2*n+2, n-k)*binomial(n+k, k). - Paul Barry, May 11 2005
Rewritten: a(n) = (1 - 2*c(n, -2))*(-1)^(n+1), n >= , with c(n, x) := Sum_{k = 0..n} C(k)*x^k and C(k) := A000108(k) (Catalan). - Wolfdieter Lang, Oct 31 2005
Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ 2^(3*n+4)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012
a(n) = hypergeometric([-n, n+1], [-n-1], 2). - Peter Luschny, Nov 30 2014
G.f.: A(x) = exp( Sum_{n >= 1} A119259(n)*x^n/n ). - Peter Bala, Jun 08 2023

A370101 a(n) = Sum_{k=0..n} binomial(4n,k) binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 7, 97, 1519, 25089, 427007, 7408897, 130287871, 2313945089, 41409732607, 745530884097, 13488086405119, 245014271688705, 4465915098890239, 81637668328243201, 1496095489290731519, 27477504726883368961, 505627095685486608383, 9320167322334416338945

Offset: 0

Seiichi Manyama, Feb 10 2024

nonn

Cf. A001448, A370100, A370102.
Cf. A119259, A370097.
Cf. A365846.

Table[Sum[Binomial[4n,k]Binomial[4n-k-1,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Dec 09 2024 *)
Table[Sum[2^k*(-1)^(n-k)*Binomial[4*n, k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 31 2025 *)

a(n) = sum(k=0, n, binomial(4*n, k)*binomial(4*n-k-1, n-k));

a(n) = [x^n] ( (1+x)^4/(1-x)^3 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^4 ). See A365846.
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k). - Seiichi Manyama, Jul 31 2025
a(n) ~ 2^(9*n + 3/2) / (7 * sqrt(Pi*n) * 3^(3*n - 1/2)). - Vaclav Kotesovec, Jul 31 2025
a(n) = Sum_{k=0..n} 2^k * binomial(3*n+k-1,k). - Seiichi Manyama, Aug 01 2025
a(n) = [x^n] 1/((1-x) * (1-2*x)^(3*n)). - Seiichi Manyama, Aug 09 2025

A116881 Row sums of triangle A116880 (generalized Catalan C(1,2)).

Original entry on oeis.org

1, 4, 23, 150, 1037, 7408, 54035, 399850, 2990105, 22540260, 170991647, 1303789534, 9983164453, 76711854040, 591236890667, 4568611684306, 35382196437041, 274564234870732, 2134337640202295, 16617270658727878

Offset: 0

Wolfdieter Lang, Mar 24 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A116880, A119259, A178792.

CoefficientList[Series[(32 x^2 + 12 Sqrt[1 - 8 x] x - 4 x) / (-32 x^3 + Sqrt[1 - 8 x] (8 x^2 + 7 x - 1) - 36 x^2 - 3 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 23 2014 *)

a(n):=sum(((k+1)^2*binomial(2*(n+1),n-k)*binomial(n+k+2,n+1))/((n+k+1)*(n+k+2)),k,0,n); /* Vladimir Kruchinin, Nov 23 2014 */

a(n) = Sum_{m=0..n} A116880(n,m), n>=0.
G.f.: (32*x^2+12*sqrt(1-8*x)*x-4*x)/(-32*x^3+sqrt(1-8*x)*(8*x^2+7*x-1)-36*x^2-3*x+1). - Vladimir Kruchinin, Nov 23 2014
a(n) = sum(k=0..n, ((k+1)^2*binomial(2*(n+1),n-k)*binomial(n+k+2,n+1))/((n+k+1)*(n+k+2))). - Vladimir Kruchinin, Nov 23 2014
a(n) ~ 2^(3*n+3) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 23 2014
Conjecture: n*(3*n-4)*a(n) +(-21*n^2+43*n-10)*a(n-1) -4*(3*n-1)*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 22 2016
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = Sum_{k=0..n} binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
a(n) = [x^n] (1+x)^(2*n+1)/(1-x)^n.
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^n).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(2*n+1,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(n+k-1,k). (End)

A156894 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2n+k-1,k).

Original entry on oeis.org

1, 3, 19, 138, 1059, 8378, 67582, 552576, 4563235, 37972290, 317894394, 2674398268, 22590697614, 191475925332, 1627653567916, 13870754053388, 118464647799075, 1013709715774130, 8689197042438274, 74594573994750972, 641252293546113434, 5519339268476249676, 47558930664216470628

Offset: 0

Paul Barry, Feb 17 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Bala, Notes on A156894

Cf. A002003, A003169, A103885, A114496, A123164.

A156894:= func< n | (&+[ Binomial(n,k)*Binomial(2*n+k-1,k): k in [0..n]]) >;
[A156894(n): n in [0..30]]; // G. C. Greubel, Jan 06 2022

a := n -> hypergeom([-n, 2*n], [1], -1);
seq(round(evalf(a(n),32)), n=0..19); # Peter Luschny, Aug 02 2014

Table[Sum[Binomial[n,k]Binomial[2n+k-1,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Nov 12 2014 *)

a(n) = if (n < 1, 1, sum(k=0, n, binomial(n,k)*binomial(2*n+k-1,k)));
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 05 2015

[round( hypergeometric([-n, 2*n], [1], -1) ) for n in (0..30)] # G. C. Greubel, Jan 06 2022

a(n) = [x^n] ((1+x)/(1-x)^2)^n.
a(n) = (4*(n+1)*(2*n+1)*A003169(n+1) - (5*n+1)*(2*n-1)*A003169(n))/(17*n + 5) for n>0. - Mark van Hoeij, Jul 14 2010
a(n) = Hypergeometric2F1([-n, 2*n], [1], -1). - Peter Luschny, Aug 02 2014
Conjecture: 64*n*(2*n-1)*a(n) -16*(89*n^2 -134*n +63)*a(n-1) +4*(661*n^2 -2619*n +2576)*a(n-2) -3*(119*n^2 -713*n +1092)*a(n-3) +6*(2*n-7)*(n-4)*a(n-4) = 0. - R. J. Mathar, Feb 05 2015
Conjecture: 16*n*(782*n +5365)*(2*n-1)*a(n) +8*(3128*n^3 -362053*n^2 +593930*n -290328)*a(n-1) -3*(726869*n^3 -5105981*n^2 +11667946*n -8715544)*a(n-2) +158*(2*n-5)*(n-3)*(391*n -764)*a(n-3) = 0. - R. J. Mathar, Feb 05 2015
Conjecture: 4*n*(2*n-1)*(17*n^2 -52*n +39)*a(n) -(1207*n^4 -4899*n^3 +6692*n^2 -3504*n +576)*a(n-1) +2*(n-2)*(2*n-3)*(17*n^2 -18*n +4)*a(n-2) = 0. - R. J. Mathar, Feb 05 2015 [the Maple command sumrecursion (binomial(n,k) * binomial(2*n+k-1,k), k, a(n)) verifies this recurrence. - Peter Bala, Oct 05 2015 ]
a(n) ~ sqrt(578 + 306*sqrt(17)) * (71 + 17*sqrt(17))^n / (17 * sqrt(Pi*n) * 2^(4*n+2)). - Vaclav Kotesovec, Feb 05 2015
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 3*x + 14*x^2 + 79*x^3 + ... is the o.g.f. of A003169 (taken with offset 0). - Peter Bala, Oct 05 2015
From Peter Bala, Mar 20 2020: (Start)
a(p) == 3 ( mod p^3 ) for prime p >= 5. Cf. A002003, A103885 and A119259.
More generally, we conjecture that a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. (End)

A333564 a(n) = [x^n] ( c(x)/c(-x) )^n, where c(x) = (1 - sqrt( 1 - 4x))/(2x) is the o.g.f. of the Catalan numbers A000108.

Original entry on oeis.org

2, 8, 56, 384, 2752, 20096, 148864, 1114112, 8403968, 63787008, 486584320, 3727196160, 28649455616, 220869853184, 1707123245056, 13223868760064, 102636144295936, 797982357192704, 6213784327684096, 48452953790480384, 378291752487878656, 2956824500391378944

Offset: 1

Peter Bala, Apr 07 2020

It can be shown that a(n) satisfies the Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) for all prime p. We conjecture that a(n) satisfies the stronger congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 3 and positive integers n and k. Some examples are given below.

More generally, for integer r and positive integer s, we conjecture that the sequence a(r,s;n) := 2^(r*n) * Sum_{k = 0..s*n-1} (-1)^(s*n-1+k)*( binomial(n+k,k) )^r satisfies the same congruences. This is the case r = s = 1.

			Examples of congruences:
a(11) - a(1) = 486584320 - 2 = 2*(11^3)*182789 == 0 ( mod 11^3 ).
a(2*11) - a(2) = 2956824500391378944 - 8 = (2^3)*(3^2)*(11^3)*
107*288357478039 == 0 ( mod 11^3 ).
a(5^2) - a(5) = 1420245922851693002752 - 2752 = (2^6)*(3^3)*(5^6)* 7*19*1123*352183001 == 0 ( mod 5^6 ).

Cf. A000108, A014300, A119259, A333565, A349648.

seq( (2^n)*add( (-1)^(n-1+k)*binomial(n+k,k), k = 0..n-1), n = 1..25);
# alternative program
a := proc (n) option remember; `if`(n = 1, 2, `if`(n = 2, 8, `if`(n = 3, 56, ((3*n+4)*a(n-1)+(36*n-76)*a(n-2)+(32*n-80)*a(n-3))/n)))
end proc:
seq(a(n), n = 1..25);

a[n_] := -2^(n) Binomial[2n, n-1] Hypergeometric2F1[1, 2n +1, n + 2, 2];
Table[Simplify[a[n]], {n, 1, 22}] (* Peter Luschny, Apr 13 2020 *)
c[x_] := (1-Sqrt[1-4x])/(2x); ser[n_] := Series[(c[x]/c[-x])^n, {x, 0, 22}];
Table[SeriesCoefficient[ser[n], n], {n, 1, 22}] (* Peter Luschny, Apr 14 2020 *)

from itertools import count, islice
def A333564_gen(): # generator of terms
    yield (a:=2)
    c = 1
    for n in count(1):
        yield a<>1
A333564_list = list(islice(A333564_gen(),20)) # Chai Wah Wu, Apr 26 2023

a(n) = 2^n * Sum_{k = 0..n-1} (-1)^(n-1+k)*binomial(n+k,k) = 2^n * A014300(n).
a(n) = (-1)^(n+1) + Sum_{k = 0..n-1} n^2/((n-k)*(2*n-k))*C(n-k,k)*C(3*n-2*k-1,n-k).
Congruences: a(p) == 2 ( mod p^3 ) for all prime p >= 3, follows from previous formula.
a(n) = Sum_{k = 1..n} (-1)^(n+k)*(3*k-1)*2^(k-1)*A000108(k-1).
a(n) = (1/2)*(A119259(n) - (-1)^n).
a(n) ~ 8^n / (3*sqrt(Pi*n)).
P-recursive with recurrence n*(3*n - 4)*a(n) = (21*n^2 - 40*n + 12)*a(n-1) + 4*(3*n - 1)*(2*n - 3)*a(n-2) with a(1) = 2 and a(2) = 8. Cf. A333565.
Alternative form: (a(n) + a(n-1))/(a(n) - a(n-2)) = P(n)/Q(n), where P(n) = 4*(3*n - 1)*(2*n - 3) and Q(n) = (21*n^2 - 40*n + 12).
Also, n*a(n) = (3*n + 4)*a(n-1) + 4*(9*n - 19)*a(n-2) + 16*(2*n - 5)*a(n-3) with a(1) = 2, a(2) = 8 and a(3) = 56.
O.g.f.: A(x) = 4*x/(1 - 8*x + (1 + 4*x)*sqrt(1 - 8*x)), which satisfies the differential equation (x + 1)*(4*x + 1)*(8*x - 1)*A'(x) + (16*x^2 - 4*x + 7)*A(x) + 2*(1 - 2*x) = 0.

A352373 a(n) = [x^n] ( 1/((1 - x)^2*(1 - x^2)) )^n for n >= 1.

Original entry on oeis.org

2, 12, 74, 484, 3252, 22260, 154352, 1080612, 7621526, 54071512, 385454940, 2758690636, 19810063392, 142662737376, 1029931873824, 7451492628260, 54013574117106, 392188079586468, 2851934621212598, 20766924805302984, 151403389181347160, 1105047483656041080

Offset: 1

Peter Bala, Mar 14 2022

Suppose n identical objects are distributed in 3*n labeled baskets, 2*n colored white and n colored black. White baskets can contain any number of objects (or be empty), while black baskets must contain an even number of objects (or be empty). a(n) is the number of distinct possible distributions.
Number of nonnegative integer solutions to n = x_1 + x_2 + ... + x_(2*n) + 2*y_1 + 2*y_2 + ... + 2*y_n.
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.
Calculation suggests that, in fact, stronger congruences may hold.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and positive integers n and k.
More generally, let r and s be integers and define a sequence (a(r,s;n))n>=1 by a(r,s;n) = [x^n] ( (1 + x)^r * (1 - x)^s )^n.
Conjecture: for each r and s the above supercongruences hold for the sequence (a(r,s;n))n>=1.
The present sequence is the case r = -1 and s = -3. Other cases include A000984 (r = 2, s = 0), A001700 with offset 1 (r = 0, s = -1), A002003 (r = 1, s = -1), A091527 (r = 3, s = -1), A119259 (r = 2, s = -1), A156894 (r = 1, s = -2), A165817 (r = 0, s = -2), A234839 (r = 1, s = 2), A348410 (r = -1, s = -2) and A351857 (r = -2, s = -4).

			n = 2: 12 distributions of 2 identical objects in 4 white and 2 black baskets
             White         Black
   1)   (0) (0) (0) (0)   [2] [0]
   2)   (0) (0) (0) (0)   [0] [2]
   3)   (2) (0) (0) (0)   [0] [0]
   4)   (0) (2) (0) (0)   [0] [0]
   5)   (0) (0) (2) (0)   [0] [0]
   6)   (0) (0) (0) (2)   [0] [0]
   7)   (1) (1) (0) (0)   [0] [0]
   8)   (1) (0) (1) (0)   [0] [0]
   9)   (1) (0) (0) (1)   [0] [0]
  10)   (0) (1) (1) (0)   [0] [0]
  11)   (0) (1) (0) (1)   [0] [0]
  12)   (0) (0) (1) (1)   [0] [0]
Examples of supercongruences:
a(7) - a(1) = 154352 - 2 = 2*(3^2)*(5^2)*(7^3) == 0 (mod 7^3);
a(2*11) - a(2) = 1105047483656041080 - 12 = (2^2)*3*(11^3)*13*101*103*2441* 209581 == 0 (mod 11^3).

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A000984, A001448, A001700, A002003, A091527, A119259, A156894, A165817, A211419, A211421, A234839, A262733, A276098, A348410, A351856, A351857.

seq(add( binomial(3*n-2*k-1,n-2*k)*binomial(n+k-1,k), k = 0..floor(n/2)), n = 1..25);

nterms=25;Table[Sum[Binomial[3n-2k-1,n-2k]Binomial[n+k-1,k],{k,0,Floor[n/2]}],{n,nterms}] (* Paolo Xausa, Apr 10 2022 *)

a(n) = Sum_{k = 0..floor(n/2)} binomial(3*n-2*k-1,n-2*k)*binomial(n+k-1,k).
a(n) = Sum_{k = 0..n} (-1)^k*binomial(4*n-k-1,n-k)*binomial(n+k-1,k).
a(n) = binomial(4*n-1,n)*hypergeom([n, -n], [1-4*n], -1).
48*n*(n-1)*(3*n-1)*(3*n-2)*(93*n^3-434*n^2+668*n-339)*a(n) = 12*(n-1)*(21762*n^6-134199*n^5+323805*n^4-386685*n^3+237728*n^2-70336*n+7680)*a(n-1) + 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(93*n^3-155*n^2+79*n-12)*a(n-2) with a(1) = 2 and a(2) = 12.
The o.g.f. A(x) = 2*x + 12*x^2 + 74*x^3 + ... is the diagonal of the bivariate rational function x*t/(1 - t/((1 - x)^2*(1 - x^2))) and hence is an algebraic function over Q(x) by Stanley 1999, Theorem 6.33, p. 197.
A(x) = x*d/dx(log(F(x))), where F(x) = (1/x)*Series_Reversion( x*(1 - x)^2*(1 - x^2) ).
a(n) ~ sqrt(4 + sqrt(6)) * (13/4 + 31*sqrt(6)/18)^n / (2*sqrt(5*Pi*n)). - Vaclav Kotesovec, Mar 15 2022

A370097 a(n) = Sum_{k=0..n} binomial(3n,k) binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 5, 49, 545, 6401, 77505, 956929, 11976193, 151388161, 1928363009, 24712450049, 318255628289, 4115300220929, 53396370030593, 694845537386497, 9064787191660545, 118516719269445633, 1552528215946035201, 20372392543502991361, 267736366910401413121

Offset: 0

Seiichi Manyama, Feb 10 2024

nonn

Cf. A091527, A370098.
Cf. A119259, A370101.
Cf. A365842.

Table[Sum[2^k*(-1)^(n-k)*Binomial[3*n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 31 2025 *)

a(n) = sum(k=0, n, binomial(3*n, k)*binomial(3*n-k-1, n-k));

a(n) = [x^n] ( (1+x)^3/(1-x)^2 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)^2/(1+x)^3 ). See A365842.
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n,k). - Seiichi Manyama, Jul 31 2025
a(n) ~ 3^(3*n + 1/2) / (5 * sqrt(Pi*n) * 2^(n-1)). - Vaclav Kotesovec, Jul 31 2025
a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k-1,k). - Seiichi Manyama, Aug 01 2025
a(n) = [x^n] 1/((1-x) * (1-2*x)^(2*n)). - Seiichi Manyama, Aug 09 2025

cp's OEIS Frontend

A064062 Generalized Catalan numbers C(2; n).

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A014300 Number of nodes of odd outdegree in all ordered rooted (planar) trees with n edges.

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A119258 Triangle read by rows: T(n,0) = T(n,n) = 1 and for 0

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A062992 Row sums of unsigned triangle A062991.

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A370101 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(4*n-k-1,n-k).

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A116881 Row sums of triangle A116880 (generalized Catalan C(1,2)).

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A370101 a(n) = Sum_{k=0..n} binomial(4n,k) binomial(4*n-k-1,n-k).

A156894 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2n+k-1,k).

A333564 a(n) = [x^n] ( c(x)/c(-x) )^n, where c(x) = (1 - sqrt( 1 - 4x))/(2x) is the o.g.f. of the Catalan numbers A000108.

A370097 a(n) = Sum_{k=0..n} binomial(3n,k) binomial(3*n-k-1,n-k).