A381145 -id:A381145 - cp's OEIS frontend

A381142 Expansion of e.g.f. exp( -LambertW(-sin(x)) ).

1, 1, 3, 15, 113, 1137, 14355, 218239, 3883585, 79218721, 1822842243, 46717337007, 1319891043569, 40759239427857, 1365932381706963, 49373610759452575, 1914856819983977473, 79316216447375396161, 3494800326874932467331, 163218136611270923087439

Offset: 0

Seiichi Manyama, Feb 15 2025

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A002017, A381145, A381148.

Cf. A136630, A185690, A277498.

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

E.g.f. A(x) satisfies A(x) = exp( sin(x) * A(x) ).

a(n) = Sum_{k=0..n} (k+1)^(k-1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A381148 E.g.f. A(x) satisfies A(x) = exp( sin(x * A(x)) / A(x) ).

Original entry on oeis.org

1, 1, 1, 0, -11, -88, -459, -560, 27945, 502336, 5223945, 18968576, -671465123, -20909349888, -345616002627, -2437013715968, 65881260463697, 3252353828442112, 76987773739473809, 873339053357432832, -17521718791602049595, -1354633521318944473088

Offset: 0

Seiichi Manyama, Feb 15 2025

sign

Cf. A002017, A381142, A381145.

Cf. A136630, A185690.

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

a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A385426 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-arcsin(x)) ).

Original entry on oeis.org

1, 1, 3, 17, 145, 1665, 24115, 422305, 8681985, 205042625, 5471351875, 162811832625, 5345929731025, 192007183247425, 7488448738333875, 315170338129570625, 14238153926819850625, 687220571240324330625, 35293921478604240911875, 1921751625123502012140625

Offset: 0

Seiichi Manyama, Jun 28 2025

nonn

Cf. A385424, A385427.

Cf. A001147, A381145, A227464, A385343.

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*exp(-asin(x)))/x))

E.g.f. A(x) satisfies A(x) = exp( arcsin(x*A(x)) ).

a(n) = Sum_{k=0..n} (n+1)^(k-1) * A385343(n,k).

cp's OEIS Frontend

A381142 Expansion of e.g.f. exp( -LambertW(-sin(x)) ).

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A381148 E.g.f. A(x) satisfies A(x) = exp( sin(x * A(x)) / A(x) ).

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A385426 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-arcsin(x)) ).

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