A364475 -id:A364475 - cp's OEIS frontend

A366221 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^2A(x)^3.

1, 1, 5, 25, 145, 905, 5941, 40433, 282721, 2018897, 14661349, 107945993, 803922289, 6045458905, 45840518933, 350100674785, 2690717983169, 20794719218593, 161502488175557, 1259855507859193, 9867012143508305, 77554946281194793, 611575725258403061

Offset: 0

Seiichi Manyama, Oct 04 2023

nonn

Cf. A364475, A366200, A366222.
Cf. A002478, A073155.
Cf. A366176, A366434.

nmax = 22; A[_] = 1;
Do[A[x_] = 1 + x*(1 + x)^2*A[x]^3 + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)

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

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

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366434.

A365178 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x).

Original entry on oeis.org

1, 1, 5, 30, 210, 1595, 12791, 106574, 913562, 8004861, 71375653, 645536234, 5907683486, 54605672300, 509043322720, 4780441915832, 45182744331388, 429472919087158, 4102806757542542, 39370967793387086, 379335734835510622, 3668220243145708341

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..988

Cf. A002294, A365180, A365181, A365182, A365183.
Cf. A364475, A365184.
Cf. A002293, A364987.

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

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(k,n-k)/(4*k+1) = Sum_{k=0..n} binomial(k,n-k) * A002293(k).

A365184 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x).

Original entry on oeis.org

1, 1, 6, 45, 395, 3775, 38146, 400826, 4335455, 47951065, 539823620, 6165377836, 71261299056, 831990025420, 9797505040130, 116235417614900, 1387958781395535, 16668362761081560, 201190667288072005, 2439418470063468505, 29698136499328762445

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..898

Cf. A002295, A365185, A365186, A365187, A365188, A365189.
Cf. A364475, A365178.
Cf. A002294, A349332.

a(n) = sum(k=0, n, binomial(k, n-k)*binomial(5*k, k)/(4*k+1));

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(k,n-k)/(5*k+1) = Sum_{k=0..n} binomial(k,n-k) * A002294(k).

A364474 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x).

Original entry on oeis.org

1, 1, 4, 16, 77, 403, 2228, 12800, 75653, 457022, 2809266, 17514200, 110480475, 703850686, 4522217364, 29268545416, 190645760149, 1248817411471, 8221323983431, 54365667330636, 360954069730636, 2405225494066647, 16080210766344354, 107828663888705292

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..1175
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 26.

Cf. A002293, A104979, A186997, A255673, A361245, A364475, A364478.

A364474 := proc(n)
    add( binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k)/(2*n-4*k+1),k=0..n/2) ;
end proc:
seq(A364474(n),n=0..80); # R. J. Mathar, Jul 27 2023

Table[Sum[Binomial[3*n - 5*k, k]*Binomial[3*n - 6*k, n - 2*k]/(2*n - 4*k + 1), {k, 0, Floor[n/2]}], {n, 0, 25}] (* Wesley Ivan Hurt, May 25 2024 *)

a(n) = sum(k=0, n\2, binomial(3*n-5*k, k)*binomial(3*n-6*k, n-2*k)/(2*n-4*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k) / (2*n-4*k+1).

D-finite with recurrence 2*n*(2*n+1)*(3*n-7)*a(n) -3*(3*n-1)*(3*n-7)*(3*n-2) *a(n-1) -2*(n-3)*(18*n^2-33*n+4) *a(n-2) +2*(18*n^3-141*n^2+287*n-64) *a(n-4) -2*(n-4)*(3*n-1)*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jul 27 2023

A371576 G.f. satisfies A(x) = ( 1 + xA(x)^(3/2) (1 + x) )^2.

Original entry on oeis.org

1, 2, 9, 44, 240, 1390, 8404, 52426, 334964, 2180928, 14418123, 96525656, 653077411, 4458529390, 30674865164, 212472058410, 1480446579602, 10369560147798, 72972217926122, 515674254743332, 3657933383804959, 26036659997517572, 185905008055923918

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Column k=2 of A378323.
Cf. A001629, A052705, A371577, A371578.
Cf. A270386, A364475.

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

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

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A364475.

A364478 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^8.

Original entry on oeis.org

1, 1, 4, 23, 154, 1124, 8675, 69626, 575243, 4859778, 41789764, 364565277, 3218581695, 28702642553, 258172627259, 2339496034381, 21337716782873, 195726876816623, 1804472496834650, 16711389876481027, 155395461519245354, 1450298253483719944

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475.

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

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

A378323 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3r+k,r) * binomial(r,n-r)/(3*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 18, 0, 1, 4, 15, 44, 94, 0, 1, 5, 22, 79, 240, 529, 0, 1, 6, 30, 124, 450, 1390, 3135, 0, 1, 7, 39, 180, 737, 2685, 8404, 19270, 0, 1, 8, 49, 248, 1115, 4532, 16585, 52426, 121732, 0, 1, 9, 60, 329, 1599, 7066, 28624, 105147, 334964, 785496, 0

Offset: 0

Seiichi Manyama, Nov 23 2024

			Square array begins:
  1,    1,    1,     1,     1,     1,     1, ...
  0,    1,    2,     3,     4,     5,     6, ...
  0,    4,    9,    15,    22,    30,    39, ...
  0,   18,   44,    79,   124,   180,   248, ...
  0,   94,  240,   450,   737,  1115,  1599, ...
  0,  529, 1390,  2685,  4532,  7066, 10440, ...
  0, 3135, 8404, 16585, 28624, 45655, 69021, ...

Columns k=0..2 give A000007, A364475, A371576.
Cf. A038137, A071943.
Cf. A378318.

T(n, k, t=3, u=0) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * (1 + x) * A_k(x)^(3/k) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A364475.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x^2 * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-2,k+2) for n > 1.

A379190 G.f. A(x) satisfies A(x) = (1 + xA(x)^3) (1 + x*A(x))^3.

Original entry on oeis.org

1, 4, 30, 304, 3557, 45150, 604222, 8393282, 119872890, 1749183075, 25964512607, 390828464403, 5951561595889, 91523131078999, 1419293428538496, 22169968253466467, 348507676062911520, 5509187208564734328, 87522347516801353980, 1396619714730284551913, 22375420057050167868366

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A379172, A379188, A379191.
Cf. A002293, A274735, A364475.
Cf. A198953.

a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(3*n+6*k+3, n-k)/(n+2*k+1));

a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(3*n+6*k+3,n-k)/(n+2*k+1).

A366222 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^4A(x)^3.

Original entry on oeis.org

1, 1, 7, 42, 287, 2114, 16338, 130802, 1075355, 9025656, 77021482, 666267502, 5829209046, 51492030953, 458612500526, 4113879873624, 37133888342707, 337041718357465, 3074153880004188, 28162578841220534, 259020296989987934, 2390818256963083305

Offset: 0

Seiichi Manyama, Oct 04 2023

nonn

Cf. A364475, A366200, A366221.

Cf. A099235, A360082.

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

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

A367040 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.

Original entry on oeis.org

1, 1, 4, 15, 70, 360, 1953, 11008, 63837, 378390, 2282205, 13960890, 86411232, 540166219, 3405341160, 21625820793, 138216775785, 888371346825, 5738510504979, 37234351046835, 242567430368298, 1585979835198675, 10403866383915844, 68453912880893025

Offset: 0

Seiichi Manyama, Nov 03 2023

nonn

Cf. A176697, A367041.
Cf. A366266, A366676.
Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475, A364478.

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

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

cp's OEIS Frontend

A366221 G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^2*A(x)^3.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A365178 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A365184 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A364474 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A371576 G.f. satisfies A(x) = ( 1 + x*A(x)^(3/2) * (1 + x) )^2.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A364478 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^8.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A378323 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3r+k,r) * binomial(r,n-r)/(3*r+k) for k > 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A379190 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * (1 + x*A(x))^3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A366222 G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^4*A(x)^3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A366221 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^2A(x)^3.

A365178 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x).

A365184 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x).

A364474 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x).

A371576 G.f. satisfies A(x) = ( 1 + xA(x)^(3/2) (1 + x) )^2.

A364478 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^8.

A379190 G.f. A(x) satisfies A(x) = (1 + xA(x)^3) (1 + x*A(x))^3.

A366222 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^4A(x)^3.