A001764 -id:A001764 - cp's OEIS frontend

A251691 G.f.: G(F(x)) is a power series in x consisting entirely of positive integer coefficients such that G(F(x) - x^k) has negative coefficients for k>0, where G(x) = 1 + x*G(x)^3 is the g.f. of A001764 and F(x) is g.f. of A251690.

1, 1, 2, 4, 8, 17, 36, 78, 169, 370, 813, 1793, 3971, 8817, 19631, 43804, 97938, 219357, 492072, 1105398, 2486320, 5598805, 12620832, 28477139, 64311189, 145354456, 328772330, 744155150, 1685434388, 3819629781, 8661130303, 19649713303, 44601771038, 101285994072, 230110466746

Offset: 0

Paul D. Hanna, Dec 31 2014

nonn

Paul D. Hanna, Table of n, a(n) for n = 0..120

Cf. A251690, A001764.

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 17*x^5 + 36*x^6 + 78*x^7 + 169*x^8 + 370*x^9 + 813*x^10 + 1793*x^11 + 3971*x^12 +...
such that A(x) = G(F(x)), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 +...
and F(x) is the g.f. of A251690:
F(x) = x - x^2 - 2*x^3 - 2*x^4 - x^6 - 3*x^8 - 3*x^10 - 3*x^11 - 3*x^13 - 2*x^14 - 3*x^15 - x^16 - 2*x^17 - x^19 - 2*x^20 - 2*x^23 - 2*x^27 - 3*x^29 +...

A381911 Expansion of (1/x) * Series_Reversion( x * (1-x) / B(x) ), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 2, 9, 55, 394, 3102, 25969, 226891, 2045342, 18883205, 177640462, 1696658418, 16408796013, 160366113609, 1581329919636, 15713344659359, 157187582466527, 1581676730708500, 15998326150898211, 162571286470135097, 1658893916098102321, 16991130941208846890

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381912, A381913.
Cf. A381817, A381914.
Cf. A001764.

Table[Sum[Binomial[n + 3*k + 1, k] * Binomial[2*n - k, n - k]/(n + 3*k + 1), {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 22 2025 *)

a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(2*n-k, n-k)/(n+3*k+1));

G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x*A(x)).

a(n) = Sum_{k=0..n} binomial(n+3*k+1,k) * binomial(2*n-k,n-k)/(n+3*k+1).

A382037 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^3) ), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 9, 160, 4325, 157896, 7280077, 406085632, 26599741065, 2001864880000, 170236619802161, 16144762562002944, 1689534516295056301, 193403842876754728960, 24040636567791329323125, 3224829927677539092791296, 464325325579881390473331473, 71428455280041816247241637888

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A052873, A382036, A382038.

Cf. A001764, A377830, A382033.

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

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * B(x*A(x))^3).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(3*n,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x)^3 ) ).

A085358 Runs of zeros in binomial(3k,k)/(2k+1) (Mod 2): relates ternary trees (A001764) to the infinite Fibonacci word (A003849).

Original entry on oeis.org

1, 2, 5, 1, 10, 1, 2, 21, 1, 2, 5, 1, 42, 1, 2, 5, 1, 10, 1, 2, 85, 1, 2, 5, 1, 10, 1, 2, 21, 1, 2, 5, 1, 170, 1, 2, 5, 1, 10, 1, 2, 21, 1, 2, 5, 1, 42, 1, 2, 5, 1, 10, 1, 2, 341, 1, 2, 5, 1, 10, 1, 2, 21, 1, 2, 5, 1, 42, 1, 2, 5, 1, 10, 1, 2, 85, 1, 2, 5, 1, 10, 1, 2, 21, 1, 2, 5, 1, 682, 1, 2, 5, 1

Offset: 0

Paul D. Hanna, Jun 25 2003

nonn

Has complementary parity to the infinite Fibonacci word: a(n) = 1 - A003849(n) (Mod 2). Records are given by A000975 and occur at Fibonacci numbers: {1,2,5,10,21,42,85,...} occur at {1,2,3,5,8,13,21,...}.

Cf. A001764 (ternary trees), A003849 (infinite Fibonacci word), A000975 (records), A085357.

Construction: start with strings S(1)={1} and S(2)={1, 2}; for k>2, let L=largest number in current string S(k); to obtain S(k+1), append S(k-1) to the end of S(k) and then replace the last number in this resulting string with {2L+1 (k odd) or 2L (k even)}. String lengths have Fibonacci growth: {1}, {1, 2}, {1, 2, 5}, {1, 2, 5, 1, 10}, {1, 2, 5, 1, 10, 1, 2, 21}, ...

A120921 G.f. satisfies: A(x) = G(x) * A(x^4G(x)^9), where G(x) is the g.f. of A001764: G(x) = 1 + xG(x)^3.

Original entry on oeis.org

1, 1, 3, 12, 56, 283, 1503, 8262, 46591, 267984, 1565949, 9269559, 55465035, 334919996, 2038268620, 12489068727, 76980573203, 476994419698, 2969444848029, 18563305700106, 116485903375761, 733457500802353

Offset: 0

Paul D. Hanna, Jul 17 2006

nonn

Self-convolution cube equals A120920, which equals column 0 of triangle A120919 (cascadence of (1+x)^3).

Cf. A120919, A120920, A001764; A001764 (ternary trees).

{a(n)=local(A=1+x,G=(1/x*serreverse(x/(1+x+x*O(x^n))^3))^(1/3)); for(i=0,n,A=G*subst(A,x,x^4*G^9 +x*O(x^n)));polcoeff(A,n,x)}

A153293 G.f.: A(x) = F(xF(x)^3) = F(F(x)-1) where F(x) = 1 + xF(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 6, 42, 317, 2508, 20517, 172180, 1474689, 12843768, 113444721, 1014062898, 9158151426, 83449247979, 766340138037, 7085966319858, 65919413472834, 616559331247512, 5794778945023698, 54700034442193302, 518375457403431600

Offset: 0

Paul D. Hanna, Jan 14 2009

nonn

			G.f.: A(x) = F(x*F(x)^3) = 1 + x + 6*x^2 + 42*x^3 + 317*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 +...

Robert Israel, Table of n, a(n) for n = 0..994

Cf. A001764, A000108; A153292, A153294.

S:= (1/2)*GAMMA(n+1/3)*GAMMA(n+2/3)*hypergeom([4/3, 5/3, -n+1], [5/2, 2*n+2], -27/4)*27^n*sqrt(3)/(Pi*GAMMA(2*n+2)):
1, seq(simplify(S),n=1..40); # Robert Israel, Dec 26 2017

F[x_] = 1 + InverseSeries[x/(1 + x)^3 + O[x]^21];
CoefficientList[F[F[x] - 1], x] (* Jean-François Alcover, Nov 02 2019 *)

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

a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(3n,n-k)*k/n for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*F(x)^3*A(x)^3 where F(x) is the g.f. of A001764.
G.f. satisfies: A(x/G(x)) = F(x*G(x)^2) = F(G(x)-1) where G(x) = F(x/G(x)) is the g.f. of A000108 and F(x) is the g.f. of A001764.
a(n) = sqrt(3)*Gamma(n+2/3)*Gamma(n+1/3)*hypergeom([4/3, 5/3, -n+1], [5/2, 2*n+2], -27/4)*27^n/(2*Pi*(n+1)!) for n >= 1. - Robert Israel, Dec 26 2017

A153299 G.f.: A(x) = F(xG(x)) where F(x) = G(xF(x)) = 1 + xF(x)^3 is the g.f. of A001764 and G(x) = F(x/G(x)) = 1 + xG(x)^2 is the g.f. of A000108 (Catalan).

Original entry on oeis.org

1, 1, 4, 20, 111, 657, 4067, 26028, 170913, 1145446, 7804797, 53911104, 376669462, 2657391772, 18904566514, 135460704648, 976795422828, 7082951967141, 51614974500605, 377798933519164, 2776363089297553, 20476554379564305

Offset: 0

Paul D. Hanna, Jan 15 2009

nonn

			G.f.: A(x) = F(x*G(x)) = 1 + x + 4*x^2 + 20*x^3 + 111*x^4 +...
Related expansions.
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 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 +...
A(x)^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 278*x^4 + 1696*x^5 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 85*x^3 + 513*x^4 + 3225*x^5 +...
G(x)*A(x)^3 = 1 + 4*x + 20*x^2 + 111*x^3 + 657*x^4 +...

Cf. A000108, A001764; A153298, A153390.

{a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(2*(n-k)+k,n-k)*k/(2*(n-k)+k)))}

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

A168479 G.f. satisfies: A(x/A(x)) = G(x)^3 where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 3, 21, 217, 2895, 46479, 857670, 17619348, 394066449, 9445681950, 239946999264, 6407385578778, 178774882463450, 5188026867995184, 156036783823130184, 4850255971984578744, 155467140310522090338

Offset: 0

Paul D. Hanna, Dec 06 2009

nonn

			G.f.: A(x) = 1 + 3*x + 21*x^2 + 217*x^3 + 2895*x^4 + 46479*x^5 +...
A(x/A(x)) = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...+ A001764(n+1)*x^n +...

Cf. A168478, A168449 (variant), A001764.

{a(n)=local(A=1+x, F=sum(k=0, n, binomial(3*k+1, k)/(3*k+1)*x^k)+x*O(x^n)); for(i=0, n, A=subst(F^3, x, serreverse(x/(A+x*O(x^n))))); polcoeff(A, n)}

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

G.f. satisfies: A(x) = [1 + A(x)*Series_Reversion(x/A(x))]^3.
G.f. satisfies: A( (x*(1-x)^2)/A(x*(1-x)^2) ) = 1/(1-x)^3.
G.f. satisfies: A( (x/(1+x)^3)/A(x/(1+x)^3) ) = (1 + x)^3.
Self-convolution cube of A168478.

A188912 Binomial convolution of the binomial coefficients bin(3n,n)/(2n+1) (A001764).

Original entry on oeis.org

1, 2, 8, 42, 260, 1816, 13962, 116094, 1029124, 9609144, 93569808, 942642696, 9763181946, 103455616400, 1117379189926, 12264816349938, 136501928050116, 1537591374945704, 17503603786398576, 201128739609458904, 2330480521265639136

Offset: 0

Emanuele Munarini, Apr 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..93

Cf. A005809, A001764, A005809, A006256, A006013, A045721, A188911, A188913

Table[Sum[Binomial[n,k]Binomial[3k,k]/(2k+1)Binomial[3n-3k,n-k]/(2n-2k+1), {k,0,n}], {n,0,22}]

makelist(sum(binomial(n,k)*binomial(3*k,k)/(2*k+1)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} binomial(n, k)*binomial(3*k, k)*binomial(3*n-3*k, n-k)/((2*k+1)*(2*n-2*k+1)).
E.g.f.: F(1/3,2/3;1,3/2;27*x/4)^2, where F(a1,a2;b1,b2;z) is a hypergeometric series.
From Vaclav Kotesovec, Jun 10 2019: (Start)
Recurrence: 8*n^2*(n+1)*(2*n+1)^2*(9*n^3-54*n^2+84*n-35)*a(n) = 24*n*(324*n^7-2187*n^6+4689*n^5-4185*n^4+1464*n^3+122*n^2-223*n+44)*a(n-1) - 18*(n-1)*(3645*n^7-30618*n^6+96066*n^5-144585*n^4+103662*n^3-21834*n^2-10860*n+4480)*a(n-2) + 2187*(n-2)^2*(n-1)*(3*n-7)*(3*n-5)*(9*n^3-27*n^2+3*n+4)*a(n-3).
a(n) ~ 3^(3*n + 1) / (Pi * n^3 * 2^(n + 1)). (End)

A188913 Binomial convolution of the binomial coefficients bin(3n,n) (A005809) and bin(3n,n)/(2n+1) (A001764).

Original entry on oeis.org

1, 4, 24, 168, 1300, 10896, 97734, 928752, 9262116, 96091440, 1029267888, 11311712352, 126921365298, 1448378629600, 16760687848890, 196237061599008, 2320532776851972, 27676644749022672, 332568471941572944, 4022574792189178080

Offset: 0

Emanuele Munarini, Apr 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..105

Cf. A005809, A001764, A006256, A006013, A045721, A188911, A188912.

Table[Sum[Binomial[n,k]Binomial[3k,k]Binomial[3n-3k,n-k]/(2n-2k+1), {k,0,n}], {n,0,22}]

makelist(sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k,0,n),n,0,12);

a(n) = sum(k=0,n,binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1));
vector(66, n, a(n-1)) /* show terms */ /* Joerg Arndt, Apr 13 2011 */

a(n) = sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k=0..n).
E.g.f.: F(1/3,2/3;1/2,1;27*x/4)*F(1/3,2/3;1,3/2;27*x/4), where F(a1,a2;b1,b2;z) is a hypergeometric series.
Recurrence: 8*n^2 * (2*n+1)^2 * (9*n^3 - 54*n^2 + 84*n - 35)*a(n) = 24*(324*n^7 - 2187*n^6 + 4689*n^5 - 4185*n^4 + 1464*n^3 + 122*n^2 - 223*n + 44)*a(n-1) - 18*(3645*n^7 - 30618*n^6 + 96066*n^5 - 144585*n^4 + 103662*n^3 - 21834*n^2 - 10860*n + 4480)*a(n-2) + 2187*(n-2)*(n-1)*(3*n-7)*(3*n-5)*(9*n^3 - 27*n^2 + 3*n + 4)*a(n-3). - Vaclav Kotesovec, Feb 25 2014
a(n) ~ 3^(3*n+1) / (Pi*n^2*2^(n+1)). - Vaclav Kotesovec, Feb 25 2014

cp's OEIS Frontend

A251691 G.f.: G(F(x)) is a power series in x consisting entirely of positive integer coefficients such that G(F(x) - x^k) has negative coefficients for k>0, where G(x) = 1 + x*G(x)^3 is the g.f. of A001764 and F(x) is g.f. of A251690.

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A381911 Expansion of (1/x) * Series_Reversion( x * (1-x) / B(x) ), where B(x) is the g.f. of A001764.

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A382037 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^3) ), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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A085358 Runs of zeros in binomial(3k,k)/(2k+1) (Mod 2): relates ternary trees (A001764) to the infinite Fibonacci word (A003849).

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A120921 G.f. satisfies: A(x) = G(x) * A(x^4*G(x)^9), where G(x) is the g.f. of A001764: G(x) = 1 + x*G(x)^3.

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A153293 G.f.: A(x) = F(x*F(x)^3) = F(F(x)-1) where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.

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A153299 G.f.: A(x) = F(x*G(x)) 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)^2 is the g.f. of A000108 (Catalan).

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A168479 G.f. satisfies: A(x/A(x)) = G(x)^3 where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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A188912 Binomial convolution of the binomial coefficients bin(3n,n)/(2n+1) (A001764).

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A120921 G.f. satisfies: A(x) = G(x) * A(x^4G(x)^9), where G(x) is the g.f. of A001764: G(x) = 1 + xG(x)^3.

A153293 G.f.: A(x) = F(xF(x)^3) = F(F(x)-1) where F(x) = 1 + xF(x)^3 is the g.f. of A001764.

A153299 G.f.: A(x) = F(xG(x)) where F(x) = G(xF(x)) = 1 + xF(x)^3 is the g.f. of A001764 and G(x) = F(x/G(x)) = 1 + xG(x)^2 is the g.f. of A000108 (Catalan).