A382920 -id:A382920 - cp's OEIS frontend

A382916 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^3 / (1-x)^2 ).

1, 1, 6, 41, 316, 2636, 23192, 211926, 1992032, 19138016, 187091252, 1855104372, 18612229836, 188601299149, 1927443803738, 19843158497163, 205602235405524, 2142401581747657, 22436439910929038, 236023405797017891, 2492914862240934612, 26426682321857813417

Offset: 0

Seiichi Manyama, Apr 08 2025

nonn

Cf. A349331, A382917.

Cf. A382920.

a(n, r=1, s=2, t=4, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1-x)^2.

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

A382918 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) / (1-x)^2 )^2.

Original entry on oeis.org

1, 2, 11, 64, 401, 2652, 18241, 129216, 936469, 6911238, 51764834, 392494366, 3006851913, 23238830982, 180974578418, 1418728452902, 11186978492689, 88668723061112, 706042492550773, 5645331629000370, 45307653034905824, 364860349786846894, 2947299389835541583

Offset: 0

Seiichi Manyama, Apr 08 2025

nonn

Cf. A006319, A382920.

Cf. A366176.

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

G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(3/2) / (1-x)^2 )^2.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).
G.f.: B(x)^2, where B(x) is the g.f. of A366176.

cp's OEIS Frontend

A382916 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^3 / (1-x)^2 ).

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