A000108 -id:A000108 - cp's OEIS frontend

A132374 Expansion of c(7x^2)/(1 - xc(7*x^2)), where c(x) is the g.f. of A000108.

1, 1, 8, 15, 120, 274, 2192, 5531, 44248, 118686, 949488, 2654646, 21237168, 61189668, 489517344, 1443039123, 11544312984, 34648845862, 277190766896, 844131474530, 6753051796240, 20813234394492, 166505875155936, 518373091849502

Offset: 0

Philippe Deléham, Nov 10 2007

Hankel transform is 7^C(n+1,2).

Series reversion of x*(1+x)*(1+2*x+8*x^2).

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

Cf. A000108, A001405, A120730, A126087, A128386, A121724, A128387, A121725.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-28*x^2))/(14*x^2 -x*(1-Sqrt(1-28*x^2))) )); // G. C. Greubel, Nov 08 2022

CoefficientList[Series[(1-Sqrt[1-28*x^2])/(14*x^2 -x*(1-Sqrt[1-28*x^2])), {x,0,40}], x] (* G. C. Greubel, Nov 08 2022 *)

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A132374(n): return sum(7^(n-k)*A120730(n,k) for k in range(n+1))
[A132374(n) for n in range(51)] # G. C. Greubel, Nov 08 2022

a(n) = Sum_{k=0..n} A120730(n,k) * 7^(n-k).
From G. C. Greubel, Nov 08 2022: (Start)
a(n) = 4*( 2*(n+1)*a(n-1) + 7*(n-2)*a(n-2) - 56*(n-2)*a(n-3) )/(n+1).
G.f.: (1 - sqrt(1 - 28*x^2))/(14*x^2 - x*(1 - sqrt(1 - 28*x^2))). (End)

A158831 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 1, 6, 54, 640, 9380, 163576, 3305484, 75915708, 1952409954, 55573310936, 1734182983962, 58863621238500, 2159006675844616, 85088103159523296, 3585740237981536700, 160894462797493581048, 7658326127259130753070

Offset: 1

Paul D. Hanna, Mar 28 2009

nonn

Triangle A158835 transforms this sequence into A158832, the next diagonal in A158825.

			Table of coefficients in the i-th iteration of x*Catalan(x):
(1);
1,(1),2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
1,2,(6),21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,12,(54),260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,110,(640),3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,1330,(9380),67844,500619,3755156,28558484,...;
1,6,42,315,2464,19852,(163576),1372196,11682348,100707972,...;
1,7,56,476,4200,38052,351792,(3305484),31478628,303208212,...;
1,8,72,684,6720,67620,693048,7209036,(75915708),807845676,...;
1,9,90,945,10230,113190,1273668,14528217,167607066,(1952409954),...; ...
where terms in parenthesis form the initial terms of this sequence.

Cf. A158825, A158832, A158833, A158834.

nmax = 18;
g[x_] := Module[{y}, Expand[Normal[(1 - Sqrt[1 - 4*y])/2 + O[y]^(nmax+2)] /. y -> x][[1 ;; nmax+1]]];
T = Table[Nest[g, x, n] // CoefficientList[#, x]& // Rest, {n, 1, nmax+1}];
Prepend[Diagonal[T, 1], 1] (* Jean-François Alcover, Jul 13 2018 *)

{a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n-1,G=subst(F,x,G));polcoeff(G,n)}

A244158 If n = Sum c_i * 10^i then a(n) = Sum c_i * Cat(i+1), where Cat(k) = A000108(k).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 5

Offset: 0

Antti Karttunen, Jun 22 2014

This sequence converts any number from various "Catalan Base number systems" (when represented as decimal numbers) back to the integer the numeral represents: e.g. we have a(A014418(n)) = n and a(A244159(n)) = n (except for the latter this is eventually broken by the shortcomings of the decimal representation used, while for the former it works for all n, because no digits larger than 3 will ever appear in the terms of A014418).
A197433 is similar, but replaces 2^k with A000108(k+1) in binary expansion of n.
For 1- and 2-digit numbers the same as A156230. - R. J. Mathar, Jun 27 2014

Index entries for sequences related to decimal expansion of n

Cf. A000108, A014418, A244159, A197433, A244155, A244156, A244157.

Differs from A028897 and A081594 for the first time at n=100, which here is a(100) = 5.

A244158 := proc(n)
    local dgs,k ;
    dgs := convert(n,base,10) ;
    add( op(k,dgs)*A000108(k),k=1..nops(dgs)) ;
end proc: # R. J. Mathar, Jan 31 2015

A330965 Array read by descending antidiagonals: A(n,k) = (1 + kn)C(n) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 10, 20, 14, 1, 5, 14, 35, 70, 42, 1, 6, 18, 50, 126, 252, 132, 1, 7, 22, 65, 182, 462, 924, 429, 1, 8, 26, 80, 238, 672, 1716, 3432, 1430, 1, 9, 30, 95, 294, 882, 2508, 6435, 12870, 4862, 1, 10, 34, 110, 350, 1092, 3300, 9438, 24310, 48620, 16796

Offset: 0

Andrew Howroyd, Jan 04 2020

			Array begins:
====================================================
n\k |   0    1    2    3     4     5     6     7
----+-----------------------------------------------
  0 |   1    1    1    1     1     1     1     1 ...
  1 |   1    2    3    4     5     6     7     8 ...
  2 |   2    6   10   14    18    22    26    30 ...
  3 |   5   20   35   50    65    80    95   110 ...
  4 |  14   70  126  182   238   294   350   406 ...
  5 |  42  252  462  672   882  1092  1302  1512 ...
  6 | 132  924 1716 2508  3300  4092  4884  5676 ...
  7 | 429 3432 6435 9438 12441 15444 18447 21450 ...
  ...

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Columns k=0..12 are A000108, A000984, A001700, A051924, A051944, A051945, A050476, A050477, A050478, A050479, A050489, A050490, A050491.

A330965[n_, k_] := CatalanNumber[n]*(k*n + 1);
Table[A330965[k, n - k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Aug 24 2025 *)

T(n, k)={(1 + k*n)*binomial(2*n,n)/(n+1)}

A(n,k) = (1 + k*n)*binomial(2*n,n)/(n+1).
A(n,k) = 2*(k*n+1)*(2*n-1)*A(n-1,k)/((n+1)*(k*n-k+1)) for n > 0.
G.f. of column k: (k - 1 - (2*k-4)*x - (k-1)*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)).

A033536 Cubes of Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 8, 125, 2744, 74088, 2299968, 78953589, 2924207000, 114933031928, 4738245926336, 203152294091656, 9000469593857728, 410006814589000000, 19129277941464384000, 911218671317138401125, 44202915427981062663000

Offset: 0

N. J. A. Sloane, Dec 10 1999

nonn

Also the number of maximum independent vertex sets in the 3(n-1)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Dec 31 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Catalan Number.
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set.

Cf. A000108, A001246.

List([0..20], n-> (Binomial(2*n, n)/(n+1))^3); # G. C. Greubel, Oct 14 2019

[Catalan(n)^3: n in [0..20]]; // Vincenzo Librandi, Nov 13 2012

seq((binomial(2*n, n)/(n+1))^3, n = 0..20); # G. C. Greubel, Oct 14 2019

Table[CatalanNumber@n^3, {n, 0, 20}] (* Vincenzo Librandi, Nov 13 2012 *)
CatalanNumber[Range[0, 20]]^3 (* Eric W. Weisstein, Dec 31 2017 *)

combinat::dyckWords::count(n)^3 $ n = 0..16; // Zerinvary Lajos, Feb 15 2007

a(n) = (binomial(2*n, n)/(n+1))^3; \\ Altug Alkan, Dec 31 2017

[catalan_number(i)^3 for i in range(0,17)] # Zerinvary Lajos, May 17 2009

[catalan_number(n)^3 for n in (0..20)] # G. C. Greubel, Oct 14 2019

From Ilya Gutkovskiy, Jan 23 2017: (Start)
O.g.f.: (1 - 3F2(-1/2,-1/2,-1/2; 1,1; 64*x))/(8*x).
E.g.f.: 3F3(1/2,1/2,1/2; 2,2,2; 64*x).
a(n) ~ 64^n/(Pi^(3/2)*n^(9/2)). (End)
From Amiram Eldar, Mar 27 2022: (Start)
a(n) = A000108(n)^3.
Sum_{n>=0} a(n)/64^n = 8 - 16*Gamma(3/4)*Gamma(7/4)/(Pi*Gamma(5/4)^2). (End)

A059346 Difference array of Catalan numbers A000108 read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 2, 3, 5, 3, 4, 6, 9, 14, 6, 9, 13, 19, 28, 42, 15, 21, 30, 43, 62, 90, 132, 36, 51, 72, 102, 145, 207, 297, 429, 91, 127, 178, 250, 352, 497, 704, 1001, 1430, 232, 323, 450, 628, 878, 1230, 1727, 2431, 3432, 4862, 603, 835, 1158, 1608, 2236, 3114

Offset: 0

N. J. A. Sloane, Jan 27 2001

			Array starts:
      1       1       2       5      14      42     132     429
      0       1       3       9      28      90     297    1001
      1       2       6      19      62     207     704    2431
      1       4      13      43     145     497    1727    6071
      3       9      30     102     352    1230    4344   15483
      6      21      72     250     878    3114   11139   40143
     15      51     178     628    2236    8025   29004  105477
     36     127     450    1608    5789   20979   76473  280221
     91     323    1158    4181   15190   55494  203748  751422
    232     835    3023   11009   40304  148254  547674 2031054
    603    2188    7986   29295  107950  399420 1483380 5527750
Triangle starts:
  1;
  0,  1;
  1,  1,  2;
  1,  2,  3,  5;
  3,  4,  6,  9, 14;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
Zhousheng Mei, Suijie Wang, Pattern Avoidance of Generalized Permutations, arXiv:1804.06265 [math.CO], 2018.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024. See p. 36.

Top row is A000108, leading diagonals give A005043, A001006, A005554.
Row sums are A106640.
Cf. A000108, A000245, A026012, A033434, A106534.

T := (n,k) -> (-1)^(n-k)*binomial(2*k,k)*hypergeom([k-n,k+1/2], [k+2], 4)/(k+1): seq(seq(simplify(T(n,k)), k=0..n), n=0..10);
# Peter Luschny, Aug 16 2012, updated May 25 2021

max = 11; t = Table[ Differences[ Table[ CatalanNumber[k], {k, 0, max}], n], {n, 0, max}]; Flatten[ Table[t[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Nov 15 2011 *)

def T(n, k) :
    if k > n : return 0
    if n == k : return binomial(2*n, n)/(n+1)
    return T(n-1, k) - T(n, k+1)
A059346 = lambda n,k: (-1)^(n-k)*T(n, k)
for n in (0..5): [A059346(n,k) for k in (0..n)] # Peter Luschny, Aug 16 2012

T(n, k) = (-1)^(n-k)*binomial(2*k,k)/(k+1)*hypergeometric([k-n, k+1/2],[k+2], 4). - Peter Luschny, Aug 16 2012

More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001

A071721 Expansion of (1+x^2C^2)C^2, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 2, 6, 18, 56, 180, 594, 2002, 6864, 23868, 83980, 298452, 1069776, 3863080, 14040810, 51325650, 188574240, 695987820, 2579248980, 9593714460, 35804293200, 134032593240, 503154100020, 1893689067348, 7144084508256

Offset: 0

N. J. A. Sloane, Jun 06 2002

nonn

a(n) = A138156(n) - 4*A138156(n-1). - Alzhekeyev Ascar M, Jul 19 2011
Apparently, for n>=1, the sum of the heights of the first and last peaks in all Dyck n-paths (in paths with one peak the height counts as both first and last). - David Scambler, Oct 05 2012
For n>=1, a(n) is the total number of nonempty subtrees over all binary trees having n+1 internal nodes.  Here, a binary tree is a full (each node has two or zero children), rooted, plane (ordered), unlabeled tree.  An empty subtree is a tree attached to the root that consists only of an external node.  a(n) = 2*A002057(n-2) + A068875(n). - Geoffrey Critzer, Sep 16 2013
From Colin Defant, Sep 15 2018: (Start)
a(n) is the number of permutations pi of [n+1] such that s(pi) avoids the patterns 132, 231, 312, and 321, where s denotes West's stack-sorting map.
a(n) is the number of permutations on [n+1] that avoid the patterns 1342, 2341, 3142, 3241, 3412, and 3421. (End)

			G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 56*x^4 + 180*x^5 + 594*x^6 + 2002*x^7 + ... - _Michael Somos_, Apr 22 2022

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See p. 32.
Boram Park and Seonjeong Park, Shellable posets arising from even subgraphs of a graph, arXiv preprint arXiv:1705.06423 [math.CO], 2017.
Seonjeong Park, Real toric manifolds and shellable posets arising from graphs, 2018.

Cf. A000108, A000245, A001622, A002421, A068875.
Row sums of triangles A319251, A319252.
gf=(1+x^2*C^2)*C^m: A000782 (m=1), this sequence (m=2), A071722 (m=3), A071723 (m=4).

a := n -> `if`(n=0, 1, 6*binomial(2*n, n-1)/(n+2));
seq(a(n), n=0..24); # Peter Luschny, Jun 28 2018

Join[{1},Table[6n CatalanNumber[n]/(n+2),{n,30}]] (* Harvey P. Dale, Jun 05 2012 *)
nn=20;t=(1-(1-4x)^(1/2))/(2x);CoefficientList[Series[D[1+x (y t -y+1)^2,y]/.y->1,{x,0,nn}],x] (* Geoffrey Critzer, Sep 16 2013 *)

{a(n) = if(n<1, n==0, 6*n*(2*n)!/(n!*(n + 1)!*(n + 2)))}; /* Michael Somos, Apr 22 2022 */

a = lambda n: n*(n+1)*hypergeometric([1-n, 2-n], [4], 1) if n>0 else 1
[simplify(a(n)) for n in range(25)] # Peter Luschny, Nov 19 2014

a(n) = 6n * (2n)! / [(n+2)n!(n+1)! ], n>0. In terms of Catalan numbers (A000108), a(n) = 6n*Cat(n)/(n+2), n>0. - Ralf Stephan, Mar 11 2004
a(n) = n*(n+1)*hypergeom([1-n, 2-n], [4], 1) for n>=1. - Peter Luschny, Nov 19 2014
D-finite with recurrence -(n+2)*(n-1)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jul 18 2017
a(n) = 2*Cat(n+1) - 2*Cat(n) = 2*A000245(n) for n>=1. - Colin Defant, Jun 27 2018
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 23/18 + 7*Pi/(27*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 43/50 - 82*sqrt(5)*log(phi)/375, where phi is the golden ratio (A001622). (End)
From Michael Somos, Apr 22 2022: (Start)
G.f.: (1 - 3*x + x^2 - (1 - x) * sqrt(1 - 4*x))/x^2.
G.f.: (2 - 2*x + x^2)/(1 - 3*x + x^2 + (1 - x)*sqrt(1 - 4*x)).
G.f.: 1 + 1/((1 - x)/(1 - sqrt(1 - 4*x)) - 1/2).
a(n) = b(n+1) - b(n) for all n in Z if b(0) = 2, b(-1) = -1, a(0) = 0, a(-1) = 3, a(-2) = -1 where b = A068875.
0 = a(n)*(+16*a(n+1) -58*a(n+2) +18*a(n+3)) +a(n+1)*(+18*a(n+1) +15*a(n+2) -13*a(n+3)) +a(n+2)*(+3*a(n+2) +a(n+3)) for all n in Z if a(0) = 0, a(-1) = 3, a(-2) = -1. (End)

A081288 a(n) is the minimal i such that A000108(i) > n.

Original entry on oeis.org

0, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Antti Karttunen, Mar 17 2003

nonn

Apart from the initial term 0, each n occurs A000245(n-1) times.

Cf. A000108, A000245, A072643, A081289, A081290. Used to compute A081291.

A081288(n) = my(i=0); while(binomial(2*i, i)/(i+1) <= n, i++); i; \\ Michel Marcus, Apr 28 2020

from sympy import catalan
def a(n):
    if n==0: return 0
    i=1
    while True:
        if catalan(i)>n: return i
        else: i+=1
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 08 2017

A126694 Expansion of g.f.: 1/(1 - 7xc(x)), where c(x) is the g.f. for A000108.

Original entry on oeis.org

1, 7, 56, 455, 3710, 30282, 247254, 2019087, 16488710, 134656130, 1099686056, 8980749862, 73342721956, 598965319960, 4891549246290, 39947649057855, 326239122661830, 2664286127154330, 21758336553841440, 177693081299126610

Offset: 0

Philippe Deléham, Feb 14 2007

The Hankel transform of this sequence is 7^n = [1, 7, 49, 343, 2401, ...] . The Hankel transform of the aerated sequence with g.f. 1/(1 - 7*x^2*c(x^2)) is also 7^n.

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 - 7*x*c(x)) (mod 2). - Peter Bala, Jul 24 2016

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

Cf. A000108, A000984, A007854, A076035, A076036, A127628, A115970.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(7*Sqrt(1-4*x) -5) )); // G. C. Greubel, May 05 2019

CoefficientList[Series[2/(-5+7*Sqrt[1-4*x]), {x, 0, 30}], x] (* G. C. Greubel, May 05 2019 *)

my(x='x+O('x^30)); Vec(2/(7*sqrt(1-4*x) -5)) \\ G. C. Greubel, May 05 2019

(2/(7*sqrt(1-4*x) -5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 05 2019

a(0) = 1, a(n) = (49*a(n-1) - 7*A000108(n-1))/6 for n >= 1.
a(n) = Sum_{k = 0..n} A106566(n,k)*7^k.
a(n) = Sum_{k = 0..n} A039599(n,k)*6^k.
a(n) ~ 5 * 7^(2*n) / 6^(n+1). - Vaclav Kotesovec, Nov 29 2021

a(16) corrected by G. C. Greubel, May 05 2019

A126983 Expansion of 1/(1+x*c(x)), c(x) the g.f. of Catalan numbers A000108.

Original entry on oeis.org

1, -1, 0, -1, -2, -6, -18, -57, -186, -622, -2120, -7338, -25724, -91144, -325878, -1174281, -4260282, -15548694, -57048048, -210295326, -778483932, -2892818244, -10786724388, -40347919626, -151355847012, -569274150156

Offset: 0

Philippe Deléham, Mar 21 2007

sign

Hankel transform is (-1)^n.

Catalan transform of A033999. - R. J. Mathar, Nov 11 2008

Fung Lam, Table of n, a(n) for n = 0..1500
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.

Cf. A000108, A000957, A039599, A064310, A106566.

[1] cat [(-1/2)^n*(1 +(&+[(-2)^k*Binomial(2*k,k)/(k+1): k in [0..n-1]])): n in [1..30]]; // G. C. Greubel, Feb 27 2019

Table[(-1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]), {n, 0, 30}] (* G. C. Greubel, Feb 27 2019 *)

{a(n) = (-1/2)^n*(1+sum(k=0,n-1, (-2)^k*binomial(2*k,k)/(k+1)))};
vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 27 2019

from itertools import count, islice
def A126983_gen(): # generator of terms
    yield from (1, -1, 0)
    a, c = 0, 1
    for n in count(1):
        yield (a:=-a-(c:=c*((n<<2)+2)//(n+2))>>1)
A126983_list = list(islice(A126983_gen(),20)) # Chai Wah Wu, Apr 27 2023

[1] + [(-1/2)^n*(1 +sum((-2)^k*catalan_number(k) for k in (0..n-1))) for n in (1..30)] # G. C. Greubel, Feb 27 2019

a(n) = (-1)^n*A064310(n).
a(n) = Sum_{k=0..n} A039599(n,k)*(-2)^k.
From Philippe Deléham, Nov 15 2009: (Start)
a(n) = Sum_{k=0..n} A106566(n,k)*(-1)^k, a(0)=1.
a(n) = -A000957(n) for n>0. (End)
Recurrence: 2*(n+2)*a(n+2) = (7*n+2)*a(n+1) + 2*(2*n+1)*a(n). - Fung Lam, May 07 2014
a(n) ~ -2^(2n)/sqrt(Pi*n^3)/9. - Fung Lam, May 07 2014

cp's OEIS Frontend

A132374 Expansion of c(7*x^2)/(1 - x*c(7*x^2)), where c(x) is the g.f. of A000108.

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A158831 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

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A244158 If n = Sum c_i * 10^i then a(n) = Sum c_i * Cat(i+1), where Cat(k) = A000108(k).

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A330965 Array read by descending antidiagonals: A(n,k) = (1 + k*n)*C(n) where C(n) = Catalan numbers (A000108).

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A033536 Cubes of Catalan numbers (A000108).

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A059346 Difference array of Catalan numbers A000108 read by antidiagonals.

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A071721 Expansion of (1+x^2*C^2)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

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A132374 Expansion of c(7x^2)/(1 - xc(7*x^2)), where c(x) is the g.f. of A000108.

A330965 Array read by descending antidiagonals: A(n,k) = (1 + kn)C(n) where C(n) = Catalan numbers (A000108).

A071721 Expansion of (1+x^2C^2)C^2, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

A126694 Expansion of g.f.: 1/(1 - 7xc(x)), where c(x) is the g.f. for A000108.