A001764 -id:A001764 - cp's OEIS frontend

A153399 G.f.: A(x) = F(xG(x)^3) where F(x) = G(x/F(x)) = 1 + xF(x)^3 is the g.f. of A001764 and G(x) = F(xG(x)) = 1 + xG(x)^4 is the g.f. of A002293.

1, 1, 6, 45, 371, 3225, 29007, 267239, 2506605, 23842644, 229369064, 2227345899, 21801617643, 214862158025, 2130226863222, 21231722675274, 212613977684254, 2138164077605865, 21585420400120710, 218677042735538547

Offset: 0

Paul D. Hanna, Jan 15 2009

nonn

			G.f.: A(x) = F(x*G(x)^3) = 1 + x + 6*x^2 + 45*x^3 + 371*x^4 +... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 +...
G(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 7084*x^5 +...
A(x)^2 = 1 + 2*x + 13*x^2 + 102*x^3 + 868*x^4 + 7732*x^5 +...
A(x)^3 = 1 + 3*x + 21*x^2 + 172*x^3 + 1509*x^4 + 13764*x^5 +...
G(x)^3*A(x)^3 = 1 + 6*x + 45*x^2 + 371*x^3 + 3225*x^4 + 29007*x^5 +...

Cf. A000108, A001764, A002293; A153398, A153291.

{a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(4*(n-k)+3*k,n-k)*3*k/(4*(n-k)+3*k)))}

a(n) = Sum_{k=0..n} C(3*k+1,k)/(3*k+1) * C(4*n-k,n-k)*3*k/(4*n-k) for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^3 where G(x) is the g.f. of A002293.
G.f. satisfies: A(x/F(x)) = F(x*F(x)^2) where F(x) is the g.f. of A001764.
G.f. satisfies: A(x/H(x)^2) = F(x*H(x)) where H(x) = 1 + x*H(x)^2 is the g.f. of A000108 (Catalan) and F(x) is the g.f. of A001764.

A192665 Floor-Sqrt transform of the numbers binomial(3n,n)/(2n+1) (A001764).

Original entry on oeis.org

1, 1, 1, 3, 7, 16, 37, 88, 207, 496, 1196, 2900, 7075, 17344, 42693, 105473, 261397, 649638, 1618527, 4041401, 10111385, 25343883, 63627940, 159982510, 402802976, 1015454569, 2562911901, 6475519561, 16377581829, 41459980288, 105047450207, 266375717828, 675980743615

Offset: 0

Emanuele Munarini, Jul 07 2011

nonn

Cf. A001764.

Table[Floor[Sqrt[Binomial[3n,n]/(2n+1)]],{n,0,100}]

makelist(floor(sqrt(binomial(3*n,n)/(2*n+1))),n,0,12);

a(n) = floor(sqrt(binomial(3*n,n)/(2*n+1))).

A218441 a(n) = A000108(n)*A001764(n).

Original entry on oeis.org

1, 1, 6, 60, 770, 11466, 188496, 3325608, 61866090, 1199333850, 24030289140, 494663027040, 10414559269296, 223487031938800, 4874879691748800, 107852781825352080, 2415945569351185530, 54714061423541554650, 1251237165698155135500, 28864572348777684057000

Offset: 0

Paul D. Hanna, Oct 28 2012

G.f. of A000108, C(x), satisfies: C(x) = 1 + x*C(x)^2;

G.f. of A001764, F(x), satisfies: F(x) = 1 + x*F(x)^3.

			G.f.: A(x) = 1 + x + 6*x^2 + 60*x^3 + 770*x^4 + 11466*x^5 + 188496*x^6 + ...

Amiram Eldar, Table of n, a(n) for n = 0..705

Cf. A000108, A001764, A086201.

a[n_] := CatalanNumber[n] * Binomial[3*n, n]/(2*n+1); Array[a, 20, 0] (* Amiram Eldar, Apr 26 2025 *)

A218441[n]:=binomial(2*n, n)/(n+1)*binomial(3*n, n)/(2*n+1)$
makelist(A218441[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

{a(n)=binomial(2*n,n)/(n+1)*binomial(3*n,n)/(2*n+1)}
for(n=0,25,print1(a(n),", "))

a(n) ~ 3^(3*n+1/2)/(2*Pi*n*(n+1)*(2*n+1)) = A086201*3^(3*n+1/2)/(n*(n+1)*(2*n+1)) (using the Stirling approximation for n!). - A.H.M. Smeets, Dec 31 2022

A251662 Dirichlet convolution of Moebius function mu(n) (A008683) with Ternary numbers A001764.

Original entry on oeis.org

1, 0, 2, 11, 54, 270, 1427, 7740, 43260, 246620, 1430714, 8414356, 50067107, 300829144, 1822766463, 11124747912, 68328754958, 422030501802, 2619631042664, 16332922043614, 102240109896265, 642312449787030, 4048514844039119, 25594403732709300, 162250238001816845, 1031147983109715120

Offset: 1

Paul D. Hanna, Jan 04 2015

nonn

			G.f.: A(x) = x + 2*x^3 + 11*x^4 + 54*x^5 + 270*x^6 + 1427*x^7 + 7740*x^8 +...
where Sum_{n>=1} A(x^n*(1-x)^(2*n)) = x - x^2:
x-x^2 = A(x*(1-x)^2) + A(x^2*(1-x)^4) + A(x^3*(1-x)^6) + A(x^4*(1-x)^8) +...

Cf. A034742, A001764, A008683.

/* Dirichlet convolution of mu(n) with Ternary numbers A001764: */
{a(n) = sumdiv(n, d, moebius(n/d) * binomial(3*(d-1), d-1)/(2*d-1))}
for(n=1, 30, print1(a(n), ", "))

/* G.f. satisfies: Sum_{n>=1} A(x^n*(1-x)^(2*n)) = x-x^2. */
{a(n)=local(A=[1, 0]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(sum(n=1, #A, subst(x*Ser(A), x, (x-2*x^2+x^3 +x*O(x^#A))^n)))[#A]); A[n]}
for(n=1, 30, print1(a(n), ", "))

G.f. A(x) satisfies: Sum_{n>=1} A((x - 2*x^2 + x^3)^n) = x - x^2.

a(n) = Sum_{d|n} Moebius(n/d) * binomial(3*(d-1), d-1)/(2*d-1).

A347953 G.f.: A(x) = 1/C(-xT(x)^3), where C(x) = 1 + xC(x)^2 is the g.f. of A000108 and T(x) = 1 + x*T(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 2, 8, 35, 171, 882, 4744, 26286, 149045, 860596, 5042968, 29913676, 179270434, 1083794310, 6601817952, 40479778395, 249646876065, 1547539929810, 9637085582640, 60260786147261, 378212395786511, 2381767469829332, 15045137488662048, 95304451461770250

Offset: 0

Alexander Burstein, Nov 02 2021

nonn

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A000108, A001764, A166135.

cx := (1-sqrt(1-4*x))/2/x ;
tx := 2/sqrt(3*x)*sin( 1/3*arcsin(sqrt(27*x/4))) ;
gf := 1/subs(x=-x*tx^3,cx) ;
taylor(%,x=0,40) ;
gfun[seriestolist](%) ; # R. J. Mathar, Jul 20 2023

CoefficientList[y/.AsymptoticSolve[y-1-x(1-y+y^2)^3/y==0,y->1,{x,0,24}][[1]],x]

seq(n) = {Vec(1/subst((1 - sqrt(1 - 4*x + O(x^2*x^n))) / (2*x), x, -serreverse(x / (1+x)^3 + O(x*x^n))))} \\ Andrew Howroyd, Nov 22 2021

G.f.: A(-x*A(x)^3) = 1/A(x).
G.f.: The series reversion of x*A(x)^3 is x*A(-x)^3.
G.f.: A(x) satisfies A(x) = 1 + x*(1 - A(x) + A(x)^2)^3/A(x).
D-finite with recurrence +4*n*(4*n-1)*(4*n+1)*a(n) +6*(-342*n^3+1233*n^2-1453*n+542)*a(n-1) +243*(n-2)*(33*n^2-123*n+112)*a(n-2) +2187*(n-3)*(3*n-4)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Jul 20 2023

A381937 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 2, 6, 35, 240, 1805, 14386, 119365, 1020136, 8918423, 79380514, 716911887, 6553219720, 60513355786, 563648995020, 5289485238552, 49963186247220, 474655663418546, 4532279676629700, 43473774550929628, 418706702628897708, 4047555977981218963

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A025227, A381787, A381940.

Cf. A001764, A346646, A365178.

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

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

a(n) = A365178(n) + A365178(n-1).

A381938 G.f. A(x) satisfies A(x) = (1 + x)^2 * B(x*A(x)), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 3, 9, 52, 380, 3066, 26304, 235314, 2170312, 20487963, 196988392, 1922327792, 18990571724, 189548947601, 1908604524752, 19364096602370, 197761735366804, 2031444188437719, 20974821788118024, 217561484977675026, 2265961977605950416, 23688432825547509283

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A366694, A367640, A381941.

Cf. A001764, A381785.

a(n) = sum(k=0, n, binomial(4*k+1, k)*binomial(2*k+2, n-k)/(4*k+1));

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(2*k+2,n-k)/(4*k+1).

a(n) = A381785(n) + A381785(n-1).

A381939 G.f. A(x) satisfies A(x) = (1 + x)^3 * B(x*A(x)), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 4, 13, 74, 568, 4872, 44576, 425936, 4199616, 42404096, 436238592, 4556085248, 48179319808, 514825553408, 5550284218368, 60296483084288, 659417378381824, 7253858445852672, 80209754567786496, 891027699137609728, 9939286070426992640, 111286739309529858048

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A366695, A381860, A381942.

Cf. A001764, A367641.

a(n) = sum(k=0, n, binomial(4*k+1, k)*binomial(3*k+3, n-k)/(4*k+1));

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(3*k+3,n-k)/(4*k+1).

a(n) = A367641(n) + A367641(n-1).

A381943 G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x)^2, where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 3, 11, 60, 425, 3426, 29619, 267738, 2497889, 23866056, 232325475, 2295889266, 22971682893, 232248775669, 2368969672183, 24348849065860, 251930963865061, 2621914660411919, 27428338267887815, 288258167672381602, 3042002859317810001, 32222429872821051817

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Partial sums of A364592.
Cf. A086616, A381867, A381945.
Cf. A001764.

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

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

A381944 G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x)^3, where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 4, 16, 89, 655, 5592, 51594, 499159, 4990821, 51140527, 534152690, 5665496618, 60854697427, 660601882734, 7235771990454, 79870211543625, 887569516968685, 9921579561050637, 111487286796322366, 1258604967618419118, 14268057344239960863, 162358119295068686098

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A162481, A366034, A381947.

Cf. A001764.

a(n) = sum(k=0, n, binomial(4*k+1, k)*binomial(n+2*k+2, n-k)/(4*k+1));

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(n+2*k+2,n-k)/(4*k+1).

cp's OEIS Frontend

A153399 G.f.: A(x) = F(x*G(x)^3) where F(x) = G(x/F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x*G(x)) = 1 + x*G(x)^4 is the g.f. of A002293.

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A192665 Floor-Sqrt transform of the numbers binomial(3*n,n)/(2*n+1) (A001764).

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Mathematica

Maxima

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A218441 a(n) = A000108(n)*A001764(n).

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A251662 Dirichlet convolution of Moebius function mu(n) (A008683) with Ternary numbers A001764.

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A347953 G.f.: A(x) = 1/C(-x*T(x)^3), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108 and T(x) = 1 + x*T(x)^3 is the g.f. of A001764.

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A381937 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A001764.

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A381938 G.f. A(x) satisfies A(x) = (1 + x)^2 * B(x*A(x)), where B(x) is the g.f. of A001764.

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A381939 G.f. A(x) satisfies A(x) = (1 + x)^3 * B(x*A(x)), where B(x) is the g.f. of A001764.

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A381943 G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x)^2, where B(x) is the g.f. of A001764.

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A153399 G.f.: A(x) = F(xG(x)^3) where F(x) = G(x/F(x)) = 1 + xF(x)^3 is the g.f. of A001764 and G(x) = F(xG(x)) = 1 + xG(x)^4 is the g.f. of A002293.

A192665 Floor-Sqrt transform of the numbers binomial(3n,n)/(2n+1) (A001764).

A347953 G.f.: A(x) = 1/C(-xT(x)^3), where C(x) = 1 + xC(x)^2 is the g.f. of A000108 and T(x) = 1 + x*T(x)^3 is the g.f. of A001764.