A378323 -id:A378323 - cp's OEIS frontend

A364475 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^3.

1, 1, 4, 18, 94, 529, 3135, 19270, 121732, 785496, 5155167, 34304706, 230923653, 1569684910, 10759159000, 74281473504, 516089542684, 3605685460750, 25316226436086, 178538289189108, 1264131169628799, 8982889404251721, 64041351551534215

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Column k=1 of A378323.

Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364478.

A364475 := proc(n)
    add( binomial(3*n-3*k,k) * binomial(3*n-4*k,n-2*k)/(2*n-2*k+1),k=0..n/2) ;
end proc:
seq(A364475(n),n=0..80); # R. J. Mathar, Jul 27 2023

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

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

D-finite with recurrence 2*n*(2*n+1)*a(n) -(5*n+1)*(3*n-2)*a(n-1) +4*(-25*n^2+75*n-59) *a(n-2) +9*(-15*n^2+69*n-80)*a(n-3) -6*(3*n-8)*(3*n-10) *a(n-4)=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.

A378318 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(n,r) * binomial(3r+k,n)/(3r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 30, 0, 1, 8, 30, 84, 170, 0, 1, 10, 48, 170, 496, 1050, 0, 1, 12, 70, 296, 1050, 3140, 6846, 0, 1, 14, 96, 470, 1920, 6846, 20832, 46374, 0, 1, 16, 126, 700, 3210, 12936, 46374, 142932, 323154, 0, 1, 18, 160, 994, 5040, 22402, 89712, 323154, 1005856, 2301618, 0

Offset: 0

Seiichi Manyama, Nov 23 2024

			Square array begins:
  1,    1,     1,     1,     1,      1,      1, ...
  0,    2,     4,     6,     8,     10,     12, ...
  0,    6,    16,    30,    48,     70,     96, ...
  0,   30,    84,   170,   296,    470,    700, ...
  0,  170,   496,  1050,  1920,   3210,   5040, ...
  0, 1050,  3140,  6846, 12936,  22402,  36492, ...
  0, 6846, 20832, 46374, 89712, 159390, 266800, ...

Columns k=0..1 give A000007, A366266.
Main diagonal gives A378378.
Cf. A266213, A378317.
Cf. A378323.

T(n, k, t=0, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*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 + 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 A366266.
B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k+2) for n > 0.

cp's OEIS Frontend

A364475 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^3.

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

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PARI

Formula

A378318 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(n,r) * binomial(3r+k,n)/(3r+k) for k > 0.

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cp's OEIS Frontend

A364475 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^3.

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A378318 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(n,r) * binomial(3*r+k,n)/(3*r+k) for k > 0.

Views

Author

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Examples

Crossrefs

Programs

PARI

Formula

A364475 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^3.

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

A378318 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(n,r) * binomial(3r+k,n)/(3r+k) for k > 0.