A001764 -id:A001764 - cp's OEIS frontend

A381912 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / B(x) ), where B(x) is the g.f. of A001764.

1, 3, 17, 124, 1038, 9470, 91586, 923542, 9608323, 102403921, 1112500651, 12275235274, 137193964646, 1549964417407, 17672282336488, 203092563108610, 2350061579393077, 27357919380212638, 320186582453226290, 3765185566095185740, 44465070300433434901, 527131055014319691537

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381911, A381913.
Cf. A381879, A381915.
Cf. A001764.

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

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

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

A381913 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, 28, 245, 2422, 25860, 291106, 3405405, 41014131, 505344113, 6341182427, 80768735045, 1041645452650, 13575670575944, 178528253213469, 2366073408348545, 31571528771106126, 423794981085407622, 5718929869862880055, 77539914280883389432, 1055790501909183080512

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381911, A381912.
Cf. A381880, A381916.
Cf. A001764.

a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(4*n-k+2, n-k)/(n+3*k+1));

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

a(n) = Sum_{k=0..n} binomial(n+3*k+1,k) * binomial(4*n-k+2,n-k)/(n+3*k+1).

A381984 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 9, 94, 1649, 40146, 1246057, 47004014, 2087644449, 106709890114, 6170322084041, 398219508589662, 28376096583546769, 2212797385807852754, 187441592012756668329, 17139223549605292448686, 1682551982313514625386817, 176505773149909540258262274, 19704960849698723062181296009

Offset: 0

Seiichi Manyama, Mar 11 2025

For each positive integer k, the sequence obtained by reducing a(n) modulo k is a periodic sequence with period dividing k. For example, modulo 6 the sequence becomes [1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, ...] with period 6. Cf. A047974. - Peter Bala, Mar 13 2025

Cf. A381985, A381986, A381987.

Cf. A001763, A001764.

seq(simplify(hypergeom([-n, 1/3, 2/3], [3/2], -27/4)), n = 0..18); # Peter Bala, Mar 13 2025

Table[HypergeometricPFQ[{-n, 1/3, 2/3}, {3/2}, -27/4], {n, 0, 20}] (* Vaclav Kotesovec, Mar 14 2025 *)

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

a(n) = n! * Sum_{k=0..n} A001764(k)/(n-k)!.
From Peter Bala, Mar 13 2025: (Start)
a(n) = hypergeom([-n, 1/3, 2/3], [3/2], -27/4).
2*(2*n + 1)*a(n) = (27*n^2 - 19*n + 4)*a(n-1) - 2*(n - 1)*(27*n - 25)*a(n-2) + 27*(n - 1)*(n - 2)*a(n-3) with a(0) = 0, a(1) = 2 and a(2) = 9. (End)
a(n) ~ 3^(3*n + 1/2) * n^(n + 1/2) / (2^(2*n + 3/2) * exp(n - 4/27) * n^(3/2)). - Vaclav Kotesovec, Mar 14 2025

A381986 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)^2), where B(x) = 1 + xB(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 17, 388, 14329, 727206, 46984729, 3689119624, 341097752657, 36302764864330, 4371463743828481, 587606216836328460, 87219196719691250185, 14168990447072685567214, 2500554381188629649979593, 476391652257266128440376336, 97447147561230881896398507553

Offset: 0

Seiichi Manyama, Mar 11 2025

nonn

Cf. A381984, A381985.

Cf. A001764, A002294, A382000.

a(n) = n!*sum(k=0, n, (2*k+1)^(n-k)*binomial(5*k+1, k)/((5*k+1)*(n-k)!));

Let F(x) be the e.g.f. of A382000. F(x) = B(x*A(x)^2) = exp( 1/3 * Sum_{k>=1} binomial(3*k,k) * (x*A(x)^2)^k/k ).

a(n) = n! * Sum_{k=0..n} (2*k+1)^(n-k) * A002294(k)/(n-k)!.

A382030 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, 3, 37, 817, 25741, 1053211, 52957297, 3157457185, 217695187801, 17036331544531, 1491702434847901, 144479729938558609, 15335923797225215653, 1770255543485671432555, 220776904683577075549801, 29582947262972619472787521, 4238424613351537181204589745, 646565304924896452410832170787

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A212722, A382029, A382031.

Cf. A001764, A382043.

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

Let F(x) be the e.g.f. of A382043. F(x) = log(A(x))/x = B(x*A(x)^2).

a(n) = n! * Sum_{k=0..n-1} (2*k+1)^(n-k-1) * binomial(n+2*k,k)/((n+2*k) * (n-k-1)!) for n > 0.

A382087 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^2) ), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 7, 106, 2525, 82536, 3436867, 174045376, 10385025849, 713599868800, 55498397386751, 4819444051348224, 462246012357060373, 48531686994029295616, 5536163290789601602875, 681824639839489261060096, 90168540044259473683829873, 12744019609725371553920876544

Offset: 0

Seiichi Manyama, Mar 15 2025

nonn

Cf. A382086, A382088, A382089.

Cf. A001764, A377832, A382037.

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

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * B(x*A(x))^2).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(2*n+k-1,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x * (1-x)^2 * exp(-x) ) ).

A124817 Interpolation of A001764(n+1) and A006629(n).

Original entry on oeis.org

1, 1, 3, 4, 12, 18, 55, 88, 273, 455, 1428, 2448, 7752, 13566, 43263, 76912, 246675, 444015, 1430715, 2601300, 8414640, 15426840, 50067108, 92431584, 300830572, 558685348, 1822766520, 3402497504, 11124755664, 20858916870, 68328754959

Offset: 0

Paul Barry, Nov 08 2006

A column of number triangle A124816.

a(n)=(1/(2(n+3)))*C(3n/2+3,n/2+1)(1+(-1)^n)+(2/(n+3))*C(3(n+1)/2,(n-1)/2)(1-(-1)^n)

A127927 G.f. A(x) satisfies: [x^(2n)] A(x)/Catalan(x)^n = A001764(n) = C(3n,n)/(2n+1) and [x^(2n+1)] A(x)/Catalan(x)^n = A001764(n+1) for n>=0, where Catalan(x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 3, 9, 31, 108, 391, 1431, 5319, 19926, 75252, 285750, 1090491, 4177774, 16060401, 61916977, 239307063, 926929746, 3597296770, 13984508500, 54448030092, 212282062488, 828673761978, 3238495227846, 12669206034339

Offset: 0

Paul D. Hanna, Feb 06 2007

nonn

Main diagonal of triangle A062745: a(n) = A062745(n,n) (see formula given in A062745 by Emeric Deutsch).

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

Cf. A062745; A001764 (ternary trees), A000108 (Catalan).

[1] cat [Binomial(2*n,n) - (-1)^(n-1)*(&+[Binomial(3*k, k)*Binomial(k-n - 1, n-2*k-1)/(2*k+1): k in [0..Floor((n-1)/2)]]): n in [1..50]]; // G. C. Greubel, Apr 30 2018

a[n_] := Binomial[2*n, n] - (-1)^(n-1)*Sum[ Binomial[3*k, k]*Binomial[k - n-1, n-1-2*k]/(2*k+1), {k, 0, Floor[(n-1)/2]}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 30 2018 *)

{a(n)=binomial(2*n,n)+(-1)^n*sum(i=0,(n-1)\2, binomial(3*i,i) *binomial(i-n-1,n-1-2*i)/(2*i+1))}

a(n) = C(2*n,n) - (-1)^(n-1)*Sum_{i=0..[(n-1)/2]} C(3*i,i)*C(i-n-1,n-1-2*i)/(2*i+1).
From Vaclav Kotesovec, May 01 2018: (Start)
Recurrence: 2*(n-1)*n*(2*n + 1)*(5*n - 6)*a(n) = (n-1)^2*(115*n^2 - 138*n + 56)*a(n-1) + 4*(n-2)*(n+1)*(2*n - 3)*(5*n - 11)*a(n-2) - 36*(n-2)*(2*n - 5)*(2*n - 3)*(5*n - 1)*a(n-3).
a(n) ~ 4^n / (phi^2 * sqrt(Pi*n)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. (End)

A153292 G.f.: A(x) = F(xF(x)^2) where F(x) = 1 + xF(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 31, 211, 1516, 11295, 86423, 675051, 5361323, 43171480, 351709926, 2894115003, 24022408477, 200918146461, 1691749323232, 14329850844625, 122028162988698, 1044131083377287, 8972696721635997, 77408293908402336

Offset: 0

Paul D. Hanna, Jan 14 2009

nonn

			G.f.: A(x) = F(x*F(x)^2) = 1 + x + 5*x^2 + 31*x^3 + 211*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, A000108; A153293.

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

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

A153295 G.f.: A(x) = F(xG(x)^2) where F(x) = G(x/F(x)) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(xG(x)) = 1 + xG(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 4, 20, 110, 638, 3828, 23515, 146972, 930869, 5958094, 38462190, 250054804, 1635421543, 10750864640, 70987129653, 470542935654, 3129729034478, 20880459397920, 139689406647522, 936832986074664, 6297064070279195

Offset: 0

Paul D. Hanna, Jan 15 2009

nonn

			G.f.: A(x) = F(x*G(x)^2) = 1 + x + 4*x^2 + 20*x^3 + 110*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
A(x)^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 276*x^4 + 1656*x^5 +...
G(x)^2*A(x)^2 = 1 + 4*x + 20*x^2 + 110*x^3 + 638*x^4 +...

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

Cf. A001764, A000108, A001006; A153294, A153296.

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

a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(3n-k,n-k)*2k/(3n-k) for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*G(x)^2*A(x)^2 where G(x) is the g.f. of A001764.
G.f. satisfies: A(x/F(x)) = F(x*F(x)) where F(x) is the g.f. of A000108 (Catalan).
From Alexander Burstein, Nov 23 2019: (Start)
G.f. satisfies: A(x) = 1 + x*G(x)^3*m(x*G(x)^3), where m(x) is the g.f. of A001006 (Motzkin numbers) and G(x) is the g.f. of A001764 (ternary trees).
G.f. satisfies: A(-x*A(x)^7) = 1/A(x). (End)

cp's OEIS Frontend

A381912 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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A381913 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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A381984 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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A381986 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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A382030 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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A382087 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^2) ), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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A124817 Interpolation of A001764(n+1) and A006629(n).

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A127927 G.f. A(x) satisfies: [x^(2n)] A(x)/Catalan(x)^n = A001764(n) = C(3n,n)/(2n+1) and [x^(2n+1)] A(x)/Catalan(x)^n = A001764(n+1) for n>=0, where Catalan(x) is the g.f. of A000108.

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A153292 G.f.: A(x) = F(x*F(x)^2) where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.

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A153295 G.f.: A(x) = F(x*G(x)^2) where F(x) = G(x/F(x)) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)) = 1 + x*G(x)^3 is the g.f. of A001764.

A381986 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)^2), where B(x) = 1 + xB(x)^3 is the g.f. of A001764.

A382030 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.

A153292 G.f.: A(x) = F(xF(x)^2) where F(x) = 1 + xF(x)^3 is the g.f. of A001764.

A153295 G.f.: A(x) = F(xG(x)^2) where F(x) = G(x/F(x)) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(xG(x)) = 1 + xG(x)^3 is the g.f. of A001764.