A001764 -id:A001764 - cp's OEIS frontend

A380637 Expansion of e.g.f. exp(xG(3x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

1, 1, 19, 703, 39313, 2959921, 280935811, 32221238239, 4336213980673, 670088514363553, 116959281939738451, 22759439305951039231, 4885844614853182749649, 1147088485458553806981073, 292394958982688921734424323, 80420728320326634679448511391

Offset: 0

Seiichi Manyama, Jan 28 2025

nonn

Cf. A001764, A380512.

Cf. A380641.

a(n) = if(n==0, 1, 3^(n-1)*(n-1)!*pollaguerre(n-1, 2*n+1, -1/3));

E.g.f.: exp( (G(3*x)-1)/3 ), where G(x) is described above.
a(n) = (n-1)! * Sum_{k=0..n-1} 3^k * binomial(3*n,k)/(n-k-1)! for n > 0.
a(n+1) = 3^n * n! * LaguerreL(n, 2*n+3, -1/3).
a(n) ~ 3^(4*n - 1/2) * n^(n-1) / (2^(2*n + 3/2) * exp(n - 1/6)). - Vaclav Kotesovec, Jan 29 2025
a(n) = (-3)^(n-1)*U(1-n, 2*(1+n), -1/3), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 29 2025
E.g.f.: exp( Series_Reversion( x/(1+3*x)^3 ) ). - Seiichi Manyama, Mar 16 2025

A380640 Expansion of e.g.f. exp(xG(2x)^2) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 9, 193, 6673, 319521, 19575001, 1461908449, 128828471073, 13086232224193, 1505486837413801, 193477959856396161, 27472294970916814129, 4271180551913140331233, 721640087945607030774393, 131656978622706616938932641, 25795404137789777215960879681, 5402020596794976601680149234049

Offset: 0

Seiichi Manyama, Jan 28 2025

nonn

Cf. A001764, A380511.
Cf. A380636, A380639.
Cf. A080893, A380643.

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

a(n) = 2 * n! * Sum_{k=0..n-1} 2^k * binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.
From Vaclav Kotesovec, Jan 29 2025: (Start)
E.g.f. A(x) satisfies x = log(A(x)) * (1 - 2*log(A(x)))^2.
a(n) ~ 3^(3*n - 3/2) * n^(n-1) / (2^(n + 1/2) * exp(n - 1/6)). (End)
a(n) = 2^(n-1)*U(1-n, 2-3*n, 1/2), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 29 2025
E.g.f.: exp( Series_Reversion( x*(1-2*x)^2 ) ). - Seiichi Manyama, Mar 16 2025

A381905 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, 8, 47, 331, 2570, 21204, 182383, 1617163, 14675783, 135643839, 1272434069, 12083390801, 115934171020, 1122129142754, 10943574296787, 107433077283767, 1060800046515405, 10528321010319417, 104972259713887665, 1050936451974803973, 10560662821468607719

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381906, A381907.

Cf. A001764, A381911.

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

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

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

A381906 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * B(x)) ), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 3, 15, 100, 787, 6848, 63583, 617350, 6191888, 63650430, 667043379, 7099806346, 76538663840, 833975952491, 9169925032189, 101616966476850, 1133736002540882, 12724529836447420, 143567856744995568, 1627454706916166076, 18526192807286106198, 211694470334287787868

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381905, A381907.

Cf. A001764, A381881.

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

G.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * B(x*A(x)).

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

A381907 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 * B(x)) ), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 4, 25, 197, 1783, 17646, 185622, 2039617, 23149542, 269367631, 3196544816, 38539697456, 470773651286, 5813914938293, 72470441063067, 910587733474165, 11521140613913305, 146659482494039073, 1876975898990490298, 24137070792680577688, 311724732112458291945

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381905, A381906.

Cf. A001764, A381882.

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

G.f. A(x) satisfies A(x) = (1 + x*A(x))^3 * B(x*A(x)).

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

A381985 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)), where B(x) = 1 + xB(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 13, 217, 5937, 223641, 10725433, 625007993, 42883208609, 3386452550689, 302545287708201, 30170153462509545, 3322052185576104049, 400328811249634307249, 52406094009429908677049, 7405663486143907784247481, 1123601498350780798756198209, 182173718779147621454796872769

Offset: 0

Seiichi Manyama, Mar 11 2025

nonn

Cf. A381984, A381986.

Cf. A001764, A002293, A346646, A364987.

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

Let F(x) be the e.g.f. of A364987. F(x) = B(x*A(x)) = exp( 1/3 * Sum_{k>=1} binomial(3*k,k) * (x*A(x))^k/k ).

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A002293(k)/(n-k)!.

A382058 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 67, 1465, 44541, 1735681, 82527439, 4632741905, 299875704697, 21989097804961, 1801520077445331, 163092373817762137, 16168084561101716725, 1741946677697976052577, 202668693570279026375671, 25324088113475137179021601, 3382305512670022948599733233, 480858973986045019386825360577

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A161629, A382059.
Cf. A161635, A382032.
Cf. A001764, A377546, A382033.

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

Let F(x) be the e.g.f. of A377546. F(x) = log(A(x))/x = B(x*A(x))^2.
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(x))^2 ) ).
a(n) = 2 * n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.

A073147 Triangle of numbers {a(n,k), n >= 0, 0<=k<=n} defined by a(0,0)=1, a(n,0)=A001764(n), a(n,n)=A006013(n), a(n,n-1)=A006629(n-1).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 12, 15, 18, 30, 55, 67, 76, 88, 143, 273, 328, 364, 400, 455, 728, 1428, 1701, 1866, 2010, 2175, 2448, 3876, 7752, 9180, 9999, 10659, 11319, 12138, 13566, 21318, 43263

Offset: 0

Paul D. Hanna, Jul 18 2002

Related to generalized Catalan numbers; in particular, C(3n,n)/(2n+1) (enumerates ternary trees and also non-crossing trees)(A001764) and sum of root degrees of all noncrossing trees on nodes on a circle (A006629).

These numbers are cardinalities of some intervals in the Tamari lattices. - F. Chapoton, Jul 15 2021

			{1}, {1,2}, {3,4,7}, {12,15,18,30}, {55,67,76,88,143}, {273,328,364,400,455,728},...

Cf. A001764, A006013, A006629, A028364.

(n, m)-th entry in triangle is Sum A001764(n-k)*A001764(k), k=0..m.

A130579 Convolution of A000108 (Catalan numbers) and A001764 (ternary trees): a(n) = Sum_{k=0..n} C(2k,k) * C(3(n-k),n-k) / [(k+1)(2(n-k)+1)].

Original entry on oeis.org

1, 2, 6, 22, 92, 423, 2087, 10856, 58765, 327877, 1872490, 10890483, 64267612, 383773529, 2314271146, 14071475748, 86165249745, 530862665988, 3288219482754, 20464419717069, 127901478759153, 802421158028657

Offset: 0

Paul D. Hanna, Jun 07 2007

nonn

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A000108, A001764.

f:= proc(n) local k; add(binomial(2*k, k)/(k+1)*binomial(3*(n-k), n-k)/(2*(n-k)+1),k=0..n) end proc:
map(f, [$0..25]); # Robert Israel, Nov 12 2024

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

G.f.: A(x) = C(x)*T(x) 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.

a(n) ~ 3^(3*n+2) / ((3^(3/2) + sqrt(11)) * sqrt(Pi) * n^(3/2) * 2^(2*n+1)). - Vaclav Kotesovec, Nov 12 2024

A153291 G.f.: A(x) = F(xF(x)) where F(x) = 1 + xF(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 4, 21, 124, 782, 5145, 34873, 241682, 1704240, 12186900, 88162753, 644058237, 4744733614, 35210349041, 262976828766, 1975324849238, 14913200362138, 113107780322778, 861417424802187, 6585224638006020, 50515048389265713

Offset: 0

Paul D. Hanna, Jan 14 2009

nonn

			G.f.: A(x) = F(x*F(x)) = 1 + x + 4*x^2 + 21*x^3 + 124*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 +...

Cf. A001764; A153292.

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

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

cp's OEIS Frontend

A380637 Expansion of e.g.f. exp(x*G(3*x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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A380640 Expansion of e.g.f. exp(x*G(2*x)^2) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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A381905 Expansion of (1/x) * Series_Reversion( x / ((1+x) * B(x)) ), where B(x) is the g.f. of A001764.

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A381906 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * B(x)) ), where B(x) is the g.f. of A001764.

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A381907 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 * B(x)) ), where B(x) is the g.f. of A001764.

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A381985 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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A382058 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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A073147 Triangle of numbers {a(n,k), n >= 0, 0<=k<=n} defined by a(0,0)=1, a(n,0)=A001764(n), a(n,n)=A006013(n), a(n,n-1)=A006629(n-1).

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A130579 Convolution of A000108 (Catalan numbers) and A001764 (ternary trees): a(n) = Sum_{k=0..n} C(2k,k) * C(3(n-k),n-k) / [(k+1)(2(n-k)+1)].

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Maple

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A153291 G.f.: A(x) = F(x*F(x)) where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.

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A380637 Expansion of e.g.f. exp(xG(3x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

A380640 Expansion of e.g.f. exp(xG(2x)^2) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

A381985 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)), where B(x) = 1 + xB(x)^3 is the g.f. of A001764.

A382058 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

A153291 G.f.: A(x) = F(xF(x)) where F(x) = 1 + xF(x)^3 is the g.f. of A001764.