A381388 -id:A381388 - cp's OEIS frontend

A381389 E.g.f. A(x) satisfies A(x) = 1/( 1 - sin(x * A(x)) )^2.

1, 2, 14, 178, 3344, 83722, 2628000, 99358810, 4398573568, 223280915090, 12788876882176, 816044058415298, 57411735641690112, 4415467258014111002, 368568207039291072512, 33186631279383615035242, 3206409506796711229521920, 330893672854541429428877602

Offset: 0

Seiichi Manyama, Feb 22 2025

nonn

Cf. A136630, A381388.

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = 2*sum(k=0, n, k!*binomial(2*n+k+2, k)/(2*n+k+2)*I^(n-k)*a136630(n, k));

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381388.
a(n) = 2 * Sum_{k=0..n} k! * binomial(2*n+k+2,k)/(2*n+k+2) * i^(n-k) * A136630(n,k), where i is the imaginary unit.
E.g.f.: (1/x) * Series_Reversion( x*(1 - sin(x))^2 ).

A381479 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x)^2 * cos(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 6, 69, 1200, 28085, 828240, 29502473, 1232606592, 59114482569, 3201204188160, 193215861134989, 12862437022076928, 936256855741871677, 73978404781917941760, 6306254322850544942865, 576881179288397985054720, 56369243043268551691136657, 5859726074013471622734938112

Offset: 0

Seiichi Manyama, Feb 24 2025

nonn

As stated in the comment of A185951, A185951(n,0) = 0^n.

Cf. A185951, A364985, A381388.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+k+1, k)/(2*n+k+1)*I^(n-k)*a185951(n, k));

a(n) = Sum_{k=0..n} k! * binomial(2*n+k+1,k)/(2*n+k+1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

E.g.f.: ( (1/x) * Series_Reversion( x*(1 - x*cos(x))^2 ) )^(1/2).

cp's OEIS Frontend

A381389 E.g.f. A(x) satisfies A(x) = 1/( 1 - sin(x * A(x)) )^2.

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