A006013 -id:A006013 - cp's OEIS frontend

A380511 Expansion of e.g.f. exp(xG(x)^2) where G(x) = 1 + xG(x)^3 is the g.f. of A001764.

1, 1, 5, 55, 961, 23141, 711421, 26631235, 1175535425, 59786520841, 3442729157461, 221413508687471, 15730688410899265, 1223574846548300845, 103417508018836074701, 9437941200860641295611, 924934291227615821904001, 96881241931552168636182545, 10801002623361396194857667365

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A251569, A380512.
Cf. A080893, A380515.
Cf. A082579, A251568, A380514.
Cf. A001764, A006013, A370054, A382058.

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

a(n) = 2 * n! * Sum_{k=0..n-1} binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-3*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x*(1-x)^2 ) ). - Seiichi Manyama, Mar 15 2025

A381751 Expansion of exp( Sum_{k>=1} binomial(8k-1,2k-1) * x^k/k ).

Original entry on oeis.org

1, 7, 252, 12866, 767460, 50005591, 3449225652, 247579862356, 18301102679444, 1383742325041292, 106516121515030768, 8319491960857739258, 657680525420544788060, 52522142073165048614002, 4230907373618147894630904, 343379827862952363210331624, 28051180121294369965012932980

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381752, A381753, A381757, A381758.

Cf. A381745.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(8*k-1, 2*k-1)*x^k/k)))

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(8*k-1,2*k-1) * a(n-k).

A381752 Expansion of exp( Sum_{k>=1} binomial(10k-1,2k-1) * x^k/k ).

Original entry on oeis.org

1, 9, 525, 44067, 4338765, 467396050, 53346810991, 6339179481480, 775994115988525, 97182642466115275, 12392633418043399130, 1603634650155295053250, 210047857493659698690575, 27795006677556725604853840, 3710220786174094422360657000, 498998879378383167317202612400

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381751, A381753, A381757, A381758.

Cf. A381746.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(10*k-1, 2*k-1)*x^k/k)))

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(10*k-1,2*k-1) * a(n-k).

A030983 Number of rooted noncrossing trees with n nodes such that root has degree 1 and the child of the root has degree at least 2.

Original entry on oeis.org

0, 3, 16, 83, 442, 2420, 13566, 77539, 450340, 2650635, 15777450, 94815732, 574518536, 3506232184, 21533144486, 132980242755, 825304177544, 5144743785545, 32199189658020, 202252227085755, 1274578959894450, 8056409137803600, 51063344718826440

Offset: 3

Emeric Deutsch

nonn

From Andrei Asinowski, May 09 2020: (Start)
With offset 0 (i.e., a(0) = 0 and a(1) = 3), a(n) is the total number of down-steps after the final up-step in all 2_1-Dyck paths of length 3*n.
A 2_1-Dyck path is a lattice path with steps U = (1, 2) and d = (1, -1) that starts at (0,0), stays (weakly) above the line y = -1, and ends at the x-axis.
For n = 2, a(2) = 16 is the total number of down-steps after the final up-step in dUddUd, dUdUdd, dUUddd, UdddUd, UddUdd, UdUddd, UUdddd (thus, 1 + 2 + 3 + 1 + 2 + 3 + 4). (End)

Andrew Howroyd, Table of n, a(n) for n = 3..200
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.
Marc Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.
Index entries for sequences related to rooted trees

Column k=1 of A102892.

Cf. A006013.

h := arcsin((3*sqrt(3)*sqrt(x))/2)/3:
gf := x*(64/9)*sin(h)^6*(1 - sin(h)^2*(8/9)): ser := series(gf, x, 32):
seq(coeff(ser, x, n), n=3..25); # Peter Luschny, Aug 08 2020
# Recurrence:
a := proc(n) option remember; if n < 4 then return 0 fi; if n = 4 then return 3 fi;
-((378*n^3 - 4536*n^2 + 18102*n - 24024)*a(n - 2) + (-1271*n^3 + 10308*n^2 - 26857*n + 22020)*a(n - 1))/(180*n^3 - 1170*n^2 + 2070*n - 1080) end:
seq(a(n), n=3..25); # Peter Luschny, Aug 08 2020

a[n_] := Binomial[3n-5, n-2]/(n-1) - 2 Binomial[3n-8, n-3]/(n-2);
a /@ Range[3, 25] (* Jean-François Alcover, Nov 03 2020, after A102892 *)

a(n)=(19*n-31)*binomial(3*n-8, n-4)/(n-1)/(2*n-3); /* Joerg Arndt, Mar 07 2013 */

concat(0, Vec((g->g^3*(3-2*g))(serreverse(x-2*x^2+x^3 + O(x^25))))) \\ Andrew Howroyd, Nov 12 2017

a(n) = (19*n - 31)*binomial(3*n - 8, n - 4)/(n - 1)/(2*n - 3).
G.f.: g^3*(3 - 2*g) where g*(1 - g)^2 = x. - Mark van Hoeij, Nov 09 2011 [That is, g = (4/3) * sin((1/3)*arcsin(sqrt(27*x/4)))^2 = x*(o.g.f. of A006013). - Petros Hadjicostas, Aug 08 2020]
From Vladimir Kruchinin, Mar 06 2013: (Start)
a(n) = binomial(3*n-5, 2*n-3)/(n-1) - 2*binomial(3*n-8, 2*n-5)/(n-2), n > 2.
a(n) = Sum_{i=1..n-3} binomial(3*i-2, 2*i-1) * binomial(3*(n-i-2), 2*(n-i-2)-1)/ (i*(n-i-2)).  (End)
a(n) ~ (76*3^(3*n - 15/2))/(4^n*sqrt(Pi)*n^(3/2)). - Peter Luschny, Aug 08 2020
D-finite with recurrence 2*(n-1)*(2*n-3)*a(n) +(-43*n^2+196*n-213)*a(n-1) +2*(62*n^2-446*n+759)*a(n-2) -12*(3*n-14)*(3*n-16)*a(n-3)=0. - R. J. Mathar, Jul 26 2022
D-finite with recurrence 2*(n-1)*(n-4)*(2*n-3)*(19*n-50)*a(n) -3*(3*n-10)*(3*n-8)*(n-3)*(19*n-31)*a(n-1)=0. - R. J. Mathar, Jul 26 2022

A071948 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.

Original entry on oeis.org

1, 1, 2, 1, 4, 7, 1, 6, 18, 30, 1, 8, 33, 88, 143, 1, 10, 52, 182, 455, 728, 1, 12, 75, 320, 1020, 2448, 3876, 1, 14, 102, 510, 1938, 5814, 13566, 21318, 1, 16, 133, 760, 3325, 11704, 33649, 76912, 120175, 1, 18, 168, 1078, 5313, 21252, 70840, 197340, 444015

Offset: 0

N. J. A. Sloane, Jun 15 2002

This is the table of h(n,k) in the notation of Carlitz (p.125). The triangle (with an offset of 1 rather than 0) enumerates two-line arrays of positive integers
............1 a_2 ... a_(n-1) a_n..........
............1 b_2 ... b_(n-1) b_n..........
such that a_i <= i (2 <= i <= n) and b_2 <= a_2 <= ... <= b_n <= a_n = k.
See A193091 and A211788 for other two-line array enumerations. - Peter Bala, Aug 02 2012

			Triangle begins
  1;
  1, 2;
  1, 4,  7;
  1, 6, 18, 30;
  1, 8, 33, 88, 143;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.
S. Dulucq, Etude combinatoire de problèmes d'énumération, d'algorithmique sur les arbres et de codage par des mots, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Row sums give A001764.
Rows are the reversals of the rows of A092276.
Cf. A193091, A211788.

T := proc(n,k) if k<=n then (n-k+1)*binomial(2*n+k+1,k)/(n+1) else 0 fi end: seq(seq(T(n,k),k=0..n),n=0..10);

t[n_, k_] /; k <= n := (n-k+1)*Binomial[2*n+k+1, k]/(n+1); t[, ] = 0; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

# Computes the first n rows of the triangle.
def A071948_triangle(n) :
    D = [0 for i in (0..n+1)]; D[1] = 1
    for i in (4..2*n+3) :
        h = i//2 - 1
        for k in (1..h) : D[k] += D[k-1]
        if i%2 == 1 : print([D[z] for z in (1..h)])
A071948_triangle(10)  # Peter Luschny, Apr 01 2012

T(n, n) = A006013(n).
T(n, k) = (n-k+1)binomial(2n+k+1, k)/(n+1) if k<=n.
Let M = the infinite square production matrix
  2, 1;
  3, 2, 1;
  4, 3, 2, 1;
  5, 4, 3, 2, 1;
  ...
The top row of M^n gives reversed terms of n-th row of triangle A071948; with leftmost terms of each row generating A006013 starting (1, 2, 7, 30, 143, ...). - Gary W. Adamson, Jul 07 2011

Edited by Emeric Deutsch, Mar 04 2004

A091665 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with 2n+1 edges and a fixed outer face of 2k edges which are invariant under a rotation of a 1/2 turn.

Original entry on oeis.org

1, 2, 2, 7, 8, 3, 30, 34, 21, 4, 143, 160, 114, 44, 5, 728, 806, 609, 308, 80, 6, 3876, 4256, 3315, 1908, 715, 132, 7, 21318, 23256, 18444, 11420, 5185, 1482, 203, 8, 120175, 130416, 104652, 67856, 34520, 12600, 2814, 296, 9, 690690, 746350, 603801, 404016, 221300, 93924, 27965, 4984, 414, 10

Offset: 1

Emeric Deutsch, Mar 03 2004

Table II in the Brown reference.

			Triangle begins:
    1;
    2,   2;
    7,   8,   3;
   30,  34,  21,  4;
  143, 160, 114, 44, 5;
  ...
The T(n,n) = n solutions correspond to a regular polygon with 2n vertices and a single diagonal joining two diametrically opposite vertices. - _Andrew Howroyd_, Mar 29 2021

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

Column 1 gives A006013, column 2 gives A046649, row sums give A000305.
Same as A046652 but with rows reversed.
Cf. A046653, A091599.

T := proc(n,k) if k<=n then k*sum((2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/(2*k-j-1)!/(n-j)!,j=k..min(n,2*k-1))/(2*n-k+1)! else 0 fi end: seq(seq(T(n,k),k=1..n),n=1..11);

t[n_, k_] := If[k <= n, k*Sum[(2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/(2*k-j-1)!/(n-j)!, {j, k, Min[n, 2*k-1]}]/(2*n-k+1)!, 0]; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 20 2014, after Maple *)

T(n,k) = {k*sum(j=k, min(n, 2*k-1), (2*j-k+1)*(j-1)!*(3*n-k-j)!/((j-k+1)!*(j-k)!*(2*k-j-1)!*(n-j)!))/(2*n-k+1)!}
for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) \\ Andrew Howroyd, Mar 29 2021

T(n, k) = k*(Sum_{j=k..min(n, 2*k-1)} (2*j-k+1)*(j-1)!*(3*n-k-j)!/((j-k+1)!*(j-k)!*(2*k-j-1)!*(n-j)!))/(2*n-k+1)! for k<=n and T(n, k)=0 for k>n.

Name clarified by Andrew Howroyd, Mar 29 2021

A109971 Inverse of Riordan array (1,x(1-x)^2), A109970.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 7, 4, 1, 0, 30, 18, 6, 1, 0, 143, 88, 33, 8, 1, 0, 728, 455, 182, 52, 10, 1, 0, 3876, 2448, 1020, 320, 75, 12, 1, 0, 21318, 13566, 5814, 1938, 510, 102, 14, 1, 0, 120175, 76912, 33649, 11704, 3325, 760, 133, 16, 1, 0, 690690, 444015, 197340

Offset: 0

Paul Barry, Jul 06 2005

Row sums are A001764. Diagonal sums are A109972. Second column is A006013. Third column is A006629.

			Rows begin
1;
0,1;
0,2,1;
0,7,4,1;
0,30,18,6,1;
0,143,88,33,8,1;
Production array begins
0, 1
0, 2, 1
0, 3, 2, 1
0, 4, 3, 2, 1
0, 5, 4, 3, 2, 1
0, 6, 5, 4, 3, 2, 1,
0, 7, 6, 5, 4, 3, 2, 1
0, 8, 7, 6, 5, 4, 3, 2, 1
0, 9, 8, 7, 6, 5, 4, 3, 2, 1
... - _Philippe Deléham_, Mar 05 2013

Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
W.-j. Woan, The Lagrange inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, example 4.

Essentially the same as A092276.

Number triangle T(0, 0)=1, T(0, k)=0, k>0, T(n, k)=(k/n)*binomial(3n-k-1, n-k) otherwise; Riordan array (1, f) where f(1-f)^2=x.
T(n, k)=sum{j=0..n, ((3j+1)/(2n+j+1))(-1)^(j-k)*C(3n, 2n+j)C(j, k)}; - Paul Barry, Oct 07 2005
T(n,k)=binomial(3n-k,n-k)*2k/(3n-k). (Paul Barry, May 18 2006)

A124305 Riordan array (1, 2sqrt(3)sin(arcsin(3sqrt(3)x/2)/3)/3).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 3, 0, 3, 0, 1, 0, 0, 7, 0, 4, 0, 1, 0, 12, 0, 12, 0, 5, 0, 1, 0, 0, 30, 0, 18, 0, 6, 0, 1, 0, 55, 0, 55, 0, 25, 0, 7, 0, 1, 0, 0, 143, 0, 88, 0, 33, 0, 8, 0, 1

Offset: 0

Paul Barry, Oct 25 2006

			Triangle begins
  1,
  0,  1,
  0,  0,  1,
  0,  1,  0,  1,
  0,  0,  2,  0,  1,
  0,  3,  0,  3,  0,  1,
  0,  0,  7,  0,  4,  0,  1,
  0, 12,  0, 12,  0,  5,  0,  1
From _Paul Barry_, Sep 28 2009: (Start)
Production matrix is
  0, 1,
  0, 0, 1,
  0, 1, 0, 1,
  0, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 (End)

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001477, A001764, A006013, A055998.

Cf. A047749 (row sums), A098746 (diagonal sums), A124304 (inverse).

A124305:= func< n,k | n eq 0 select 1 else (1/2)*(1+(-1)^(n-k))*(k/n)*Binomial(n + Floor((n-k)/2) -1, n-1) >;
[A124305(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 25 2023

A124305[n_, k_]:= If[n==0, 1, (1/2)*(1+(-1)^(n-k))*(k/n)*Binomial[n +(n-k)/2 -1, (n-k)/2]];
Table[A124305[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 19 2023 *)

def A124305(n,k): return 1 if n==0 else ((n-k+1)%2)*k*binomial(n + (n-k)//2 -1, n-1)//n
flatten([[A124305(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 25 2023

Sum_{k=0..n} T(n, k) = A047749(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*(1 + (-1)^n)*A098746(n/2).
From G. C. Greubel, Aug 19 2023: (Start)
T(n, k) = (1/2)*(1 + (-1)^(n-k))*(k/n)*binomial(n + (n-k)/2 - 1, (n-k)/2), with T(0, 0) = 1.
T(n, n) = 1.
T(n, n-2) = A001477(n-2).
T(n, n-4) = A055998(n-4).
T(n, n-6) = A111396(n-6).
T(n, 0) = 0^n.
T(n, 1) = ((1-(-1)^n)/2)*A001764(floor((n-1)/2)).
T(n, 2) = ((1+(-1)^n)/2)*A006013(floor((n-2)/2)).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A047749(n). (End)

A143603 Triangle, read by rows, such that the g.f. of column k = G(x)^(2k+1) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764 (ternary trees).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 12, 12, 5, 1, 55, 55, 25, 7, 1, 273, 273, 130, 42, 9, 1, 1428, 1428, 700, 245, 63, 11, 1, 7752, 7752, 3876, 1428, 408, 88, 13, 1, 43263, 43263, 21945, 8379, 2565, 627, 117, 15, 1, 246675, 246675, 126500, 49588, 15939, 4235, 910, 150, 17, 1

Offset: 1

Paul D. Hanna, Aug 29 2008

From Peter Bala, Aug 07 2014: (Start)
Riordan array (G(x), x*G(x)). Let C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + ... be the o.g.f. of the Catalan numbers A000108. Then C(x*G(x)) = G(x).
This leads to a factorization of this array in the group of Riordan matrices as (1, x*G(x))*(C(x), x*C(x)) = (1 + A110616)*A033184 (here, in the final product, 1 refers to the 1 X 1 identity matrix and + means direct sum - see the Example section). (End)

			Triangle begins:
1;
1, 1;
3, 3, 1;
12, 12, 5, 1;
55, 55, 25, 7, 1;
273, 273, 130, 42, 9, 1;
1428, 1428, 700, 245, 63, 11, 1;
7752, 7752, 3876, 1428, 408, 88, 13, 1; ...
where g.f. of column k = G(x)^(2k+1) where G(x) = 1 + x*G(x)^3.
Matrix inverse begins:
1;
-1, 1;
0, -3, 1;
0, 3, -5, 1;
0, -1, 10, -7, 1;
0, 0, -10, 21, -9, 1;
0, 0, 5, -35, 36, -11, 1;
0, 0, -1, 35, -84, 55, -13, 1; ...
where g.f. of column k = (1-x)^(2k+1) for k>=0.
From _Peter Bala_, Aug 07 2014: (Start)
Matrix factorization as (1 + A110616)*A033184 begins
/1           \/ 1         \    / 1           \
|0  1        || 1  1       |   | 1  1        |
|0  1 1      || 2  2 1     | = | 3  3  1     |
|0  3 2 1    || 5  5 3 1   |   |12 12  5 1   |
|0 12 7 3 1  ||14 14 9 4 1 |   |55 55 25 7 1 |
(End)

Yuxuan Zhang and Yan Zhuang, A subfamily of skew Dyck paths related to k-ary trees, arXiv:2306.15778 [math.CO], 2023.

Cf. columns: A001764, A102893, A102594; row sums: A006013. A033184, A110616.

{T(n,k)=binomial(3*n-k,n-k)*(2*k+1)/(2*n+1)}

T(n,k) = C(3n-k,n-k)*(2k+1)/(2n+1) for 0<=k<=n.
Let M = the production matrix:
1, 1
2, 2, 1
3, 3, 2, 1
4, 4, 3, 2, 1
5, 5, 4, 3, 2, 1
...
Top row of M^(n-1) gives n-th row. - Gary W. Adamson, Jul 07 2011

A230547 a(n) = 3binomial(3n+9, n)/(n+3).

Original entry on oeis.org

1, 9, 63, 408, 2565, 15939, 98670, 610740, 3786588, 23535820, 146710476, 917263152, 5752004349, 36174046743, 228124619100, 1442387942520, 9142452842985, 58083251802345, 369816259792035, 2359448984037600

Offset: 0

Tim Fulford, Oct 23 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=3, r=9.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Cf. A000108, A001764, A006013, A006629, A102893, A006630, A102594, A006631, A233657.

[9*Binomial(3*n+9, n)/(3*n+9): n in [0..30]];

Table[9 Binomial[3 n + 9, n]/(3 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(3*n+9,n)/(3*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=3, r=9.

D-finite with recurrence 2*n*(2*n+9)*(n+4)*a(n) -3*(3*n+7)*(n+2)*(3*n+8)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

cp's OEIS Frontend

A380511 Expansion of e.g.f. exp(x*G(x)^2) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A381751 Expansion of exp( Sum_{k>=1} binomial(8*k-1,2*k-1) * x^k/k ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A381752 Expansion of exp( Sum_{k>=1} binomial(10*k-1,2*k-1) * x^k/k ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A030983 Number of rooted noncrossing trees with n nodes such that root has degree 1 and the child of the root has degree at least 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A071948 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A091665 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with 2*n+1 edges and a fixed outer face of 2*k edges which are invariant under a rotation of a 1/2 turn.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A109971 Inverse of Riordan array (1,x(1-x)^2), A109970.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A124305 Riordan array (1, 2*sqrt(3)*sin(arcsin(3*sqrt(3)*x/2)/3)/3).

Views

Author

Keywords

Examples

A380511 Expansion of e.g.f. exp(xG(x)^2) where G(x) = 1 + xG(x)^3 is the g.f. of A001764.

A381751 Expansion of exp( Sum_{k>=1} binomial(8k-1,2k-1) * x^k/k ).

A381752 Expansion of exp( Sum_{k>=1} binomial(10k-1,2k-1) * x^k/k ).

A091665 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with 2n+1 edges and a fixed outer face of 2k edges which are invariant under a rotation of a 1/2 turn.

A124305 Riordan array (1, 2sqrt(3)sin(arcsin(3sqrt(3)x/2)/3)/3).

A230547 a(n) = 3binomial(3n+9, n)/(n+3).