A381146 -id:A381146 - cp's OEIS frontend

A381141 Expansion of e.g.f. exp( -LambertW(-x * cos(x)) ).

1, 1, 3, 13, 89, 821, 9667, 137817, 2306705, 44308009, 960645251, 23205700453, 618086944873, 17996847978461, 568729575572355, 19387150575025201, 709130794848586657, 27704208465508996945, 1151379111946617111043, 50721472225191792506301, 2360928161776701549045241

Offset: 0

Seiichi Manyama, Feb 15 2025

nonn

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

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

Cf. A009189, A381144, A381146.

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

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

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

A381144 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * cos(x)) ).

Original entry on oeis.org

1, 1, 3, 13, 65, 221, -2933, -120903, -3104127, -71637191, -1562635789, -31373685947, -505087300991, -1692007785259, 402032879446395, 28152810613025521, 1423083552938781697, 62552808878706976625, 2459148829654813484131, 82692880516086149155581

Offset: 0

Seiichi Manyama, Feb 15 2025

sign

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

Index entries for reversions of series

Cf. A009189, A381141, A381146.

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)*I^(n-k)*a185951(n, k));

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

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

cp's OEIS Frontend

A381141 Expansion of e.g.f. exp( -LambertW(-x * cos(x)) ).

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