A047749 -id:A047749 - cp's OEIS frontend

A386380 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(6k) a(n-1-6*k).

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 15, 24, 34, 45, 57, 70, 154, 253, 368, 500, 650, 819, 1827, 3045, 4495, 6200, 8184, 10472, 23562, 39627, 59052, 82251, 109668, 141778, 320866, 543004, 814506, 1142295, 1533939, 1997688, 4540200, 7718340, 11633440, 16398200, 22137570

Offset: 0

Seiichi Manyama, Jul 20 2025

nonn

Wikipedia, Fuss-Catalan number

Cf. A002296, A233832, A233833, A386392, A233834, A130565.

Cf. A047749, A118968, A124753, A386379, A386396.

A386380 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        add(procname(6*k)*procname(n-1-6*k),k=0..floor((n-1)/6)) ;
    end if;
end proc:
seq(A386380(n),n=0..80) ; # R. J. Mathar, Jul 30 2025

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\6, 7, n%6+1);

For k=0..5, a(6*n+k) = (k+1) * binomial(7*n+k+1,n)/(7*n+k+1).

G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..5} A(w^k*x)), where w = exp(Pi*i/3).

A047755 a(n) = A047752(12n+5).

Original entry on oeis.org

0, 0, 1, 5, 26, 133, 708, 3861, 21604, 123266, 715221, 4206956, 25032840, 150413348, 911379384, 5562367173, 34164355848, 211015212580, 1309815397995, 8166460799805, 51120054233490, 321156223592865, 2024257417812240, 12797201858645100

Offset: 0

N. J. A. Sloane

nonn

Also A047754(4n+1). The proof is elementary and involves reducing both expressions via the various linear expansions with A001764, A047753, A047751 and A047749 which are found in these sequences. - R. J. Mathar, Oct 18 2008

Edited by N. J. A. Sloane, Jan 17 2009 at the suggestion of R. J. Mathar

A082938 Number of solid 2-trees with 2n+1 edges.

Original entry on oeis.org

1, 1, 1, 2, 5, 13, 49, 201, 940, 4643, 24037, 127859, 696365, 3858759, 21704863, 123619126, 711787259, 4137614454, 24256010068, 143271593982, 852001881614, 5097719884665, 30670572676389, 185466705697057

Offset: 0

N. J. A. Sloane, May 26 2003

nonn

Also, the number of noncrossing partitions up to rotation and reflection composed of n blocks of size 3. - Andrew Howroyd, May 03 2018

Andrew Howroyd, Table of n, a(n) for n = 0..200
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
M. Bousquet and C. Lamathe, Enumeration of solid trees according to edge number and edge degree distribution, Discr. Math., 298 (2005), 115-141.

Column k=3 of A303929.

Cf. A047749, A054423.

u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));
e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]
c[n_, k_] := If[n == 0, 1, (DivisorSum[n, EulerPhi[n/#]*Binomial[k*#, #] &] + DivisorSum[GCD[n-1, k], EulerPhi[#]*Binomial[n*k/#, (n-1)/#] &])/(k*n) - Binomial[k*n, n]/(n*(k - 1) + 1)];
T[n_, k_] := (1/2)*(c[n, k] + If[n == 0, 1, If[OddQ[k], If[OddQ[n], 2*u[ Quotient[n, 2], k, (k + 1)/2], u[n/2, k, 1] + u[n/2 - 1, k, k]], e[n, k] + If[OddQ[n], u[Quotient[n, 2], k, k/2]]]/2]) /. Null -> 0;
a[n_] := T[n, 3];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd and A303929 *)

a(n) = (A047749(n)+A054423(n))/2. - Vladeta Jovovic, Sep 11 2004

a(n) ~ 3^(3*n - 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 3)). - Vaclav Kotesovec, Jun 01 2022

More terms from Vladeta Jovovic, Sep 11 2004

A084081 Sum of lists created by n substitutions k -> Range[k+1,0,-2] starting with {0}, counting down from k+1 to 0 step -2.

Original entry on oeis.org

0, 1, 2, 5, 10, 24, 50, 121, 260, 637, 1400, 3468, 7752, 19380, 43890, 110561, 253000, 641355, 1480050, 3771885, 8765250, 22439040, 52451256, 134796060, 316663760, 816540124, 1926501200, 4982228488, 11798983280, 30593078076, 72690164850

Offset: 0

Wouter Meeussen, May 11 2003

nonn

Lengths of lists is A047749.

			Lists {0}, {1}, {2, 0}, {3, 1, 1}, {4, 2, 0, 2, 0, 2, 0} sum to 0, 1, 2, 5, 10.

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

Cf. A047749, A093951.

F:=Floor; B:=Binomial;
function A084081(n)
  if (n mod 2) eq 0 then return 10*B(F((3*n+2)/2), F((n-2)/2))/(n+3);
  else return 2*(3*n+1)*B(F((3*n+5)/2), F((n+1)/2))/((n+3)*(3*n+5));
  end if; return A084081;
end function;
[A084081(n): n in [0..40]]; // G. C. Greubel, Oct 17 2022

Plus@@@Flatten/@NestList[ # /. k_Integer :> Range[k+1, 0, -2]&, {0}, 8]
A084081[n_]:= If[EvenQ[n], 10*Binomial[(3*n+2)/2, (n-2)/2]/(n+3), 2*(3*n + 1)*Binomial[(3*n+5)/2, (n+1)/2]/((n+3)*(3*n+5))];
Table[A084081[n], {n, 40}] (* G. C. Greubel, Oct 17 2022 *)

def A084081(n):
    if (n%2==0): return 10*binomial(int((3*n+2)/2), int((n-2)/2))/(n+3)
    else: return 2*(3*n+1)*binomial(int((3*n+5)/2), int((n+1)/2))/((n+3)*(3*n+5))
[A084081(n) for n in range(40)] # G. C. Greubel, Oct 17 2022

Equals A093951(n) - A047749(n).
From G. C. Greubel, Oct 17 2022: (Start)
a(2*n+1) = (3*n-1)*binomial[3*n+1, n]/((n+1)*(3*n+1)).
a(2*n) = 10*binomial(3*n+1, n-1)/(2*n+3). (End)

A093951 Sum of integers generated by n-1 substitutions, starting with 1, k -> k+1, k-1, .., 1.

Original entry on oeis.org

1, 2, 4, 8, 17, 36, 80, 176, 403, 910, 2128, 4896, 11628, 27132, 65208, 153824, 373175, 888030, 2170740, 5202600, 12797265, 30853680, 76292736, 184863168, 459162452, 1117370696, 2786017120, 6804995008, 17024247304, 41717833740, 104673837384

Offset: 1

Wouter Meeussen, Apr 18 2004

nonn

Substitutions 1 -> {2}, 2 -> {3,1}, 3 -> {4,2}, 4 -> {5,3,1}, 5 -> {6,4,2}, 6 -> {7,5,3,1}, 7 -> {8,6,4,2}, etc. The function f(n) gives determinant of (I_{n} - x * A(n)) where I_{n} is the identity matrix and A(n) = 0 if j > i + 1 otherwise (i+j) mod 2, for i = 1, ..., n and j = 1, ..., n, and can be written in terms of Dickson polynomials as g(w) = x*D_(w-1)(1+x, x*(1+x)) + (1-2*x)*E_(w-1)(1+x, x*(1+x)). - Francisco Salinas (franciscodesalinas(AT)hotmail.com), Apr 13 2004
Count of integers is A047749.
Sum of integers with substitution starting from 0 is A084081.

			GF(12) = (1 + 2*x - 7*x^2 - 14*x^3 + 9*x^4 + 20*x^5 + 2*x^6 - 2*x^7 + 2*x^11)/(1 - 11*x^2 + 36*x^4 - 35*x^6 + 5*x^8) produces a(1) to a(12).
a(4)=8 since 4-1 = 3 substitutions on 1 produce 1 -> 2 -> 3+1 -> 4 + 2 + 2 = 8.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A006629, A047749, A056096, A084081.

function A093951(n)
  if (n mod 2) eq 0 then return 8*Binomial(Floor(3*n/2), Floor((n-2)/2))/(n+2);
  else return 6*Binomial(Floor((3*n+1)/2), Floor((n-1)/2))/(n+2) - 2*Binomial(Floor((3*n-1)/2), Floor((n-1)/2))/(n+1);
  end if; return A093951;
end function;
[A093951(n): n in [1..40]]; // G. C. Greubel, Oct 17 2022

Plus@@@Flatten/@NestList[ #/.k_Integer:>Range[k+1, 1, -2]&, {1}, 8];(*or for n>16 *); f[1]=1; f[2]=1-x^2; f[3]=1-2x^2; f[n_]:=f[n]=Expand[f[n-1]-x^2 f[n-3]]; g[1]=1; g[2]=1+2x; g[3]=1+2x+2x^2; g[n_]:=g[n]=Expand[g[n-1] -x^2 g[n-3]+2 x^(n-1)]; GF[n_]:=g[n]/f[n]; CoefficientList[Series[GF[36], {x, 0, 36}], x]

{a(n)=if(n%2==0,4*binomial(3*n/2,n/2-1)/(n/2+1), 6*binomial(3*(n\2)+2, n\2)/(2*(n\2)+3) - binomial(3*(n\2)+1,n\2)/(n\2+1))} \\ Paul D. Hanna, Apr 24 2006

def A093951(n):
    if (n%2==0): return 8*binomial(3*n/2, (n-2)/2)/(n+2)
    else: return 6*binomial((3*n+1)/2, (n-1)/2)/(n+2) - 2*binomial((3*n-1)/2, (n-1)/2)/(n+1)
[A093951(n) for n in range(1,40)] # G. C. Greubel, Oct 17 2022

a(n) = [x^n] GF(n) with GF(n) = g(n)/f(n) and f(1)=1, f(2)=1-x^2, f(3)=1-2*x^2, f(n) = f(n-1) - x^2*f(n-3) and g(1)=1, g(2)=1+2*x, g(3)=1+2*x+2*x^2, g(n) = g(n-1) - x^2*g(n-3) + 2*x^(n-1).
From Paul D. Hanna, Apr 24 2006: (Start)
a(2*n) = 4*binomial(3*n, n-1)/(n+1) = 2*A006629(n-1).
a(2*n+1) = 6*binomial(3*n+2, n)/(2*n+3) - binomial(3*n+1, n)/(n+1) = A056096(n+3). (End)

A219795 Sum of the absolute values of the antidiagonals of the triangle A135929(n) companion. See the comment.

Original entry on oeis.org

2, 2, 2, 2, 3, 3, 5, 7, 10, 11, 16, 23, 33, 44, 58, 81, 114, 158, 212, 293, 407, 565, 777, 1064, 1471, 2036, 2813, 3863, 5334, 7370, 10183, 14046, 19356, 26726, 36909, 50955, 70251, 96977, 133886, 184841, 255092

Offset: 0

Paul Curtz, Nov 28 2012

nonn

The companion to A135929(n) is the triangle
2;
2, 0;
2, 0,  1;
2, 0, -1, 0;
2, 0, -3, 0, -1;
2, 0, -5, 0,  0, 0;
2, 0, -7, 0,  3, 0, 1;
2, 0, -9, 0,  8, 0, 1, 0;
(A192011(n) beginning with 2 instead of -1).
Consider a(1),a(5),a(10),a(14), that is, a(A193910(n) -1).
a(1)+a(4)-a(5) = 2, a(5)+a(8)-a(9) = 2, a(10)+a(13)-a(14) = 2, a(14)+a(17)-a(18) = 4, a(19)+a(22)-a(23) = 6, a(23)+a(26)-a(27) = 14, yields 2,2,2,4,6,14,24,60,... = 2*A047749(n) or 2, followed with A116637(n+1).

			a(0)=2, a(1)=2, a(2)=2+0, a(3)=2+0, a(4)=2+0+1, a(5)=2+0+1.

A219795 := proc(n)
    if n=0 then
        2;
    else
        add(abs(A192011(n-k,k)),k=0..floor(n/2)) ;
    end if;
end proc: # R. J. Mathar, Jan 06 2013

a(n) = sum abs ( [k=0..floor(n/2)] A192011(n-k,k) ), a(0)=2.

a(24)-a(40) from Jean-Francois Alcover, Nov 28 2012

A124747 Inverse of number triangle A124744.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 2, 2, 2, 1, 0, 0, 3, 3, 3, 2, 1, 0, 0, 7, 7, 7, 5, 3, 1, 0, 0, 12, 12, 12, 9, 6, 3, 1, 0, 0, 30, 30, 30, 23, 16, 9, 4, 1, 0, 0, 55, 55, 55, 43, 31, 19, 10, 4, 1

Offset: 0

Paul Barry, Nov 06 2006

Third column and row sums give A047749.

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 0, 1, 1,
0, 0, 1, 1, 1,
0, 0, 2, 2, 2, 1,
0, 0, 3, 3, 3, 2, 1,
0, 0, 7, 7, 7, 5, 3, 1,
0, 0, 12, 12, 12, 9, 6, 3, 1

A124816 Product of Riordan array (1,x(1-x^2))^(-1) and number triangle T(n,k)=C(floor(k/2),n-k).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 0, 3, 3, 3, 2, 1, 0, 0, 7, 4, 5, 2, 1, 0, 12, 12, 12, 10, 6, 3, 1, 0, 0, 30, 18, 24, 12, 9, 3, 1, 0, 55, 55, 55, 50, 32, 22, 10, 4, 1, 0, 0, 143, 88, 121, 66, 57, 25, 14, 4, 1, 0, 273, 273, 273

Offset: 0

Paul Barry, Nov 08 2006

Product of A124305 and A124788. Columns include aerated A001764,A047749(n+1),A124817,A084081. Row sums are A124818.

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 0, 2, 1, 1,
0, 3, 3, 3, 2, 1,
0, 0, 7, 4, 5, 2, 1,
0, 12, 12, 12, 10, 6, 3, 1,
0, 0, 30, 18, 24, 12, 9, 3, 1,
0, 55, 55, 55, 50, 32, 22, 10, 4, 1

Cf. A124790.

A192894 Number of symmetric 13-ary factorizations of the n-cycle (1,2...n).

Original entry on oeis.org

1, 1, 1, 7, 13, 112, 247, 2310, 5525, 53998, 135408, 1360289, 3518515, 36017352, 95223414, 988172368, 2655417765, 27844071255, 75769712590, 801012669457, 2201663313200, 23428926096576, 64924369564353, 694644371065372, 1938034271677595, 20829931845958872, 58448142042957576

Offset: 0

N. J. A. Sloane, Jul 12 2011

nonn

The six sequences displayed in Table 1 of the Bousquet-Lamathe reference are A047749, A143546, A143547, A143554, A192893, A192894. From this one should be able to guess a g.f.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1.

Column k=13 of A369929 and k=14 of A370062.

Cf. A143049.

From Seiichi Manyama, Jul 07 2025: (Start)
G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x)*A(-x))^6 ).
G.f. A(x) satisfies A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2), where G(x) = 1 + x*G(x)^13.
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_7>=0 and x_1+2*(x_2+x_3+...+x_7)=n-1} a(x_1) * Product_{k=2..7} a(2*x_k). (End)
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_13>=0 and x_1+x_2+...+x_13=n-1} (-1)^(x_1+x_2+x_3+x_4+x_5+x_6) * Product_{k=1..13} a(x_k). - Seiichi Manyama, Jul 09 2025

a(11) onwards from Andrew Howroyd, Jan 26 2024

a(0)=1 prepended by Seiichi Manyama, Jul 07 2025

A217138 G.f. A(x) satisfies A(x) = 1 + xA(x)^2(A(x) + A(-x))/2.

Original entry on oeis.org

1, 1, 2, 7, 22, 94, 340, 1579, 6118, 29746, 120060, 600934, 2492028, 12725756, 53798888, 278786739, 1195684230, 6265816042, 27175425004, 143671870034, 628705751828, 3347680236132, 14756641134872, 79039468217086, 350529497005532, 1886818634445044, 8410852483002200

Offset: 0

Paul D. Hanna, Sep 27 2012

nonn

Compare to: G(x) = 1 + x*G(x)*(G(x) + G(-x))/2, which is the g.f. of A047749.
The radius of convergence r of g.f. A(x) is
r = 0.192450089729875254836382926833985818549200... with
A(r)  = (3 - sqrt(9-6*sqrt(2)))/sqrt(2) = 1.614014407382354328773...
A(-r) = (sqrt(9+6*sqrt(2)) - 3)/sqrt(2) = 0.835475335400823769423...
where y = A(r) and y = A(-r) solves y^4 = 18*(1-y)^2.
In closed form, r = 1/(3*sqrt(3)). - Vaclav Kotesovec, Feb 17 2014

			A(x) = 1 + x + 2*x^2 + 7*x^3 + 22*x^4 + 94*x^5 + 340*x^6 + 1579*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 62*x^4 + 260*x^5 + 1005*x^6 + 4522*x^7 +...
(A(x) + A(-x))/2 = 1 + 2*x^2 + 22*x^4 + 340*x^6 + 6118*x^8 +...

Cf. A047749.

{a(n)=local(A=1+O(x^(n+1))); for(i=0, n, A=1+x*A^2*(A+subst(A, x, -x))/2); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies: 2*(A(x)-1)^3 + x*(A(x)-1)*(2-A(x))*A(x)^3 - x^2*A(x)^6 = 0.
The formal inverse of g.f. A(x) is (x-1)*(2-x + sqrt(x^2 + 4*x - 4))/(2*x^3).
Recurrence: (n-1)*n*(n+1)*(2*n-1)*(2*n+1)*(198*n^5 - 1890*n^4 + 6915*n^3 - 12015*n^2 + 9832*n - 3060)*a(n) = 30*(n-1)*n*(2*n - 1)*(3*n - 7)*(36*n^4 - 246*n^3 + 567*n^2 - 468*n + 55)*a(n-1) + 3*(n-1)*(14256*n^9 - 193104*n^8 + 1111626*n^7 - 3526794*n^6 + 6652659*n^5 - 7420161*n^4 + 4436059*n^3 - 971921*n^2 - 90480*n - 23100)*a(n-2) - 90*(2*n - 1)*(3*n - 8)*(3*n - 7)*(3*n - 4)*(36*n^4 - 246*n^3 + 567*n^2 - 468*n + 55)*a(n-3) - 36*(n-3)*(3*n - 11)*(3*n - 10)*(3*n - 8)*(3*n - 7)*(198*n^5 - 900*n^4 + 1335*n^3 - 630*n^2 - 23*n - 20)*a(n-4). - Vaclav Kotesovec, Feb 17 2014
a(n) ~ (1+2*sqrt(2)-sqrt(3) + (-1)^n*(1-2*sqrt(2)+sqrt(3))) * 3^(3*n/2) / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 17 2014
a(0) = 1; a(n) = Sum_{i, j, k>=0 and i+j+2*k=n-1} a(i) * a(j) * a(2*k). - Seiichi Manyama, Jul 07 2025

cp's OEIS Frontend

A386380 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(6*k) * a(n-1-6*k).

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Maple

PARI

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A047755 a(n) = A047752(12n+5).

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A082938 Number of solid 2-trees with 2n+1 edges.

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Mathematica

Formula

Extensions

A084081 Sum of lists created by n substitutions k -> Range[k+1,0,-2] starting with {0}, counting down from k+1 to 0 step -2.

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Magma

Mathematica

SageMath

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A093951 Sum of integers generated by n-1 substitutions, starting with 1, k -> k+1, k-1, .., 1.

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Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A219795 Sum of the absolute values of the antidiagonals of the triangle A135929(n) companion. See the comment.

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Author

Keywords

Comments

Examples

Programs

Maple

Formula

Extensions

A124747 Inverse of number triangle A124744.

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Comments

Examples

A124816 Product of Riordan array (1,x(1-x^2))^(-1) and number triangle T(n,k)=C(floor(k/2),n-k).

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Author

Keywords

Comments

Examples

Crossrefs

A192894 Number of symmetric 13-ary factorizations of the n-cycle (1,2...n).

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A386380 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(6k) a(n-1-6*k).

A217138 G.f. A(x) satisfies A(x) = 1 + xA(x)^2(A(x) + A(-x))/2.