A002293 -id:A002293 - cp's OEIS frontend

A381915 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / B(x) ), where B(x) is the g.f. of A002293.

1, 3, 18, 145, 1378, 14515, 163700, 1936414, 23716654, 298216851, 3827542585, 49938733635, 660366743580, 8830549084588, 119205253249287, 1622258295003714, 22232669093660250, 306569446979862205, 4250285556933578693, 59210418891925845529, 828417259759216617257

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381914, A381916.
Cf. A381879, A381912.
Cf. A002293.

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

A381987 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 11, 160, 3941, 134486, 5851327, 309520436, 19283504585, 1382980764106, 112223497464371, 10165461405056552, 1016801830348902061, 111312715288354681310, 13237965546409421546471, 1699516550894276788156156, 234263144339070269872076177, 34507561203827621878485498386

Offset: 0

Seiichi Manyama, Mar 12 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 5 the sequence becomes [1, 2, 1, 0, 1, 1, 2, 1, 0, 1, ...] with period 5. In particular, a(5*n+3) == 0 (mod 5). Cf. A047974. - Peter Bala, Mar 13 2025

Cf. A381984, A381988, A381989.

Cf. A002293, A365340.

seq(simplify(hypergeom([-n, 1/2, 1/4, 3/4], [2/3, 4/3], -256/27)), n = 0..17); # Peter Bala, Mar 13 2025

Table[HypergeometricPFQ[{-n, 1/2, 1/4, 3/4}, {2/3, 4/3}, -256/27], {n, 0, 20}] (* Vaclav Kotesovec, Mar 14 2025 *)

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

a(n) = n! * Sum_{k=0..n} A002293(k)/(n-k)!.
From Peter Bala, Mar 13 2025: (Start)
a(n) = hypergeom([-n, 1/2, 1/4, 3/4], [2/3, 4/3], -256/27).
3*(3*n - 1)*(3*n + 1)*a(n) = n*(256*n^2 - 303*n + 95)*a(n-1) - 3*(n - 1)*(256*n^2 - 485*n + 245)*a(n-2) + 3*(256*n - 375)*(n - 1)*(n - 2)*a(n-3) - 256*(n - 1)*(n - 2)*(n - 3)*a(n-4) with a(0) = 1, a(1) = 2, a(2) = 11 and a(3) = 160. (End)
a(n) ~ 2^(8*n+1) * n^(n-1) / (3^(3*n + 3/2) * exp(n - 27/256)). - Vaclav Kotesovec, Mar 14 2025

A381989 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)^2), where B(x) = 1 + xB(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 19, 514, 22621, 1369546, 105616639, 9901346554, 1093292035609, 138977379784882, 19990424969236171, 3209995501651871890, 569216406245186726965, 110476637766622355475898, 23294266811686640511534199, 5302371488162151660366545866, 1295920217231693678343467474353

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A381987, A381988.

Cf. A002293, A002295, A382001.

a(n) = n!*sum(k=0, n, (2*k+1)^(n-k)*binomial(6*k+1, k)/((6*k+1)*(n-k)!));

Let F(x) be the e.g.f. of A382001. F(x) = B(x*A(x)^2) = exp( 1/4 * Sum_{k>=1} binomial(4*k,k) * (x*A(x)^2)^k/k ).

a(n) = n! * Sum_{k=0..n} (2*k+1)^(n-k) * A002295(k)/(n-k)!.

A382088 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^3) ), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 9, 178, 5549, 237456, 12945037, 858203872, 67035559257, 6029839290880, 613862192499281, 69777500840918784, 8760124051527691141, 1203852634738613966848, 179746834136205848167125, 28975042890917781500747776, 5015346425440407318539964593, 927775677566572703009955053568

Offset: 0

Seiichi Manyama, Mar 15 2025

nonn

Cf. A382086, A382087, A382089.

Cf. A002293, A377833, A382038.

a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (n+1)^(n-k-1)*binomial(3*n+k-1, 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-1,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x * (1-x)^3 * exp(-x) ) ).

A153395 G.f.: A(x) = F(xG(x)) where F(x) = G(x/F(x)^2) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(xG(x)^2) = 1 + xG(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 3, 13, 69, 417, 2754, 19373, 142732, 1088875, 8533278, 68308641, 556242792, 4593529882, 38380159009, 323860968709, 2756019889146, 23625552635184, 203823793118268, 1768357487401595, 15418860927887232, 135042445950316514

Offset: 0

Paul D. Hanna, Jan 15 2009

nonn

This appears to be the same as the sequence in row 1 of Fig. 21 of Novelli-Thibon 2014. - N. J. A. Sloane, Jun 14 2014

			G.f.: A(x) = F(x*G(x)) = 1 + x + 3*x^2 + 13*x^3 + 69*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 + 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 + 7*x^2 + 32*x^3 + 173*x^4 + 1050*x^5 +...
G(x)*A(x)^2 = 1 + 3*x + 13*x^2 + 69*x^3 + 417*x^4 + 2754*x^5 +...

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014.

Cf. A000108, A001764, A002293; A153394, A153396.

nmax = 21;
G[_] = 0;
Do[G[x_] = 1 + x*G[x]^4 + O[x]^nmax, nmax];
F[x_] = Sum[CatalanNumber[n] x^n, {n, 0, nmax}];
A[x_] = F[x G[x]];
CoefficientList[A[x], x] (* Jean-François Alcover, Sep 09 2018 *)

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

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

A153397 G.f.: A(x) = F(xG(x)^4) = F(G(x)-1) where F(x) = G(x/F(x)^2) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(xG(x)^2) = 1 + xG(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 6, 43, 334, 2717, 22776, 195000, 1695874, 14927990, 132673398, 1188412986, 10714602196, 97133633788, 884716464592, 8091061578807, 74259516900390, 683694381314696, 6312247839166260, 58424001667319720, 541971167468786770

Offset: 0

Paul D. Hanna, Jan 15 2009

nonn

			G.f.: A(x) = F(x*G(x)^4) = 1 + x + 6*x^2 + 43*x^3 + 334*x^4 +...
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 + 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 + 98*x^3 + 790*x^4 + 6618*x^5 +...
G(x)^4*A(x)^2 = 1 + 6*x + 43*x^2 + 334*x^3 + 2717*x^4 + 22776*x^5 +...

Cf. A000108, A001764, A002293; A153396, A153398.

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

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

A153398 G.f.: A(x) = F(xG(x)^2) 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.

Original entry on oeis.org

1, 1, 5, 33, 245, 1941, 16023, 136075, 1179833, 10392981, 92701411, 835271032, 7589337123, 69444928453, 639280878401, 5915683250220, 54991636090761, 513257729193329, 4807619948647095, 45177320023095160, 425766248463523359

Offset: 0

Paul D. Hanna, Jan 15 2009

nonn

			G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 33*x^3 + 245*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 + 11*x^2 + 76*x^3 + 581*x^4 + 4702*x^5 +...
A(x)^3 = 1 + 3*x + 18*x^2 + 130*x^3 + 1023*x^4 + 8457*x^5 +...
G(x)^2*A(x)^3 = 1 + 5*x + 33*x^2 + 245*x^3 + 1941*x^4 + 16023*x^5 +...

Cf. A000108, A001764, A002293; A153397, A153399.

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

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

A381908 Expansion of (1/x) * Series_Reversion( x / ((1+x) * B(x)) ), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 2, 9, 64, 556, 5351, 54818, 585941, 6459430, 72902748, 838174008, 9781930978, 115579403512, 1379879992445, 16620303073607, 201717610488447, 2464502123154530, 30286289207099652, 374115157763376043, 4642636869759251879, 57852132860181652189, 723592983110972398779

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381909, A381910.

Cf. A002293.

a(n) = sum(k=0, n, binomial(n+4*k+1, k)*binomial(n+1, n-k)/(n+4*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+4*k+1,k) * binomial(n+1,n-k)/(n+4*k+1).
a(n) = hypergeom([(1+n)/4, (2+n)/4, (3+n)/4, (4+n)/4, -n], [2, (2+n)/3, (3+n)/3, (4+n)/3], -2^8/3^3). - Stefano Spezia, Mar 10 2025

A381909 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * B(x)) ), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 3, 16, 121, 1117, 11569, 128648, 1500054, 18091859, 223794730, 2823369749, 36185653049, 469808971400, 6165903108879, 81667617713170, 1090234962290114, 14654059445570507, 198151602861222385, 2693625234657193038, 36789566028850640226, 504600217464088999466

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381908, A381910.

Cf. A002293.

a(n) = sum(k=0, n, binomial(n+4*k+1, k)*binomial(2*n+2, n-k)/(n+4*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+4*k+1,k) * binomial(2*n+2,n-k)/(n+4*k+1).
a(n) = binomial(2*(1 + n), n)*hypergeom([(1+n)/4, (2+n)/4, (3+n)/4, (4+n)/4, -n], [(2+n)/3, (3+n)/3, (4+n)/3, 3+n], -2^8/3^3)/(1 + n). - Stefano Spezia, Mar 10 2025

A381910 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 * B(x)) ), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 4, 26, 222, 2243, 25243, 305217, 3878731, 51097713, 691596081, 9558970897, 134347855874, 1914131985782, 27582542400252, 401284140631911, 5886072268606617, 86951528919335670, 1292467847124221832, 19316795168721092789, 290107272994659617741, 4375905051887803660504

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381908, A381909.

Cf. A002293.

a(n) = sum(k=0, n, binomial(n+4*k+1, k)*binomial(3*n+3, n-k)/(n+4*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+4*k+1,k) * binomial(3*n+3,n-k)/(n+4*k+1).
a(n) = binomial(3*(1 + n), n)*hypergeom([(1+n)/4, (2+n)/4, (3+n)/4, (4+n)/4, -n], [(2+n)/3, (3+n)/3, (4+n)/3, 4+2*n], -2^8/3^3)/(1 + n). - Stefano Spezia, Mar 10 2025

cp's OEIS Frontend

A381915 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / B(x) ), where B(x) is the g.f. of A002293.

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A381987 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A381989 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)^2), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A382088 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^3) ), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A153395 G.f.: A(x) = F(x*G(x)) where F(x) = G(x/F(x)^2) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)^2) = 1 + x*G(x)^4 is the g.f. of A002293.

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A153397 G.f.: A(x) = F(x*G(x)^4) = F(G(x)-1) where F(x) = G(x/F(x)^2) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)^2) = 1 + x*G(x)^4 is the g.f. of A002293.

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A153398 G.f.: A(x) = F(x*G(x)^2) 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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A381908 Expansion of (1/x) * Series_Reversion( x / ((1+x) * B(x)) ), where B(x) is the g.f. of A002293.

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A381909 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * B(x)) ), where B(x) is the g.f. of A002293.

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A381989 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)^2), where B(x) = 1 + xB(x)^4 is the g.f. of A002293.

A153395 G.f.: A(x) = F(xG(x)) where F(x) = G(x/F(x)^2) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(xG(x)^2) = 1 + xG(x)^4 is the g.f. of A002293.

A153397 G.f.: A(x) = F(xG(x)^4) = F(G(x)-1) where F(x) = G(x/F(x)^2) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(xG(x)^2) = 1 + xG(x)^4 is the g.f. of A002293.

A153398 G.f.: A(x) = F(xG(x)^2) 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.