A381140 -id:A381140 - cp's OEIS frontend

A381143 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * cosh(x)) ).

1, 1, 3, 19, 185, 2381, 38227, 739271, 16752465, 435437209, 12772234211, 417396070235, 15040805940745, 592531894182437, 25336144876513395, 1168670193628654351, 57845446906144852769, 3058248577410499021361, 172007282950136451003331, 10255035157348348977955619

Offset: 0

Seiichi Manyama, Feb 15 2025

nonn

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

Index entries for reversions of series

Cf. A003727, A162649, A381140.

Cf. A185951.

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, (n+1)^(k-1)*a185951(n, k));

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

a(n) = Sum_{k=0..n} (n+1)^(k-1) * A185951(n,k).

A381260 Expansion of e.g.f. exp( -LambertW(-2 * x * cosh(x)) / 2 ).

Original entry on oeis.org

1, 1, 5, 52, 789, 16116, 415633, 12963952, 474761577, 19974663568, 949570545501, 50341473508800, 2944936306028797, 188445333393978688, 13094108333345047401, 981850811606401705216, 79025876659261666454481, 6795576727642475440025856, 621789675800920271144983477

Offset: 0

Seiichi Manyama, Feb 18 2025

nonn

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

Cf. A185951, A381140.

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, (2*k+1)^(k-1)*a185951(n, k));

E.g.f. A(x) satisfies A(x) = exp( x * cosh(x) * A(x)^2 ).

a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * A185951(n,k).

cp's OEIS Frontend

A381143 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * cosh(x)) ).

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