A381344 -id:A381344 - cp's OEIS frontend

A381345 Expansion of e.g.f. 1/( 1 - x * cos(sqrt(2)*x) ).

1, 1, 2, 0, -24, -220, -1200, -2576, 52864, 1016208, 10909440, 57039488, -687971328, -26190716864, -450123634688, -4238375059200, 24514848522240, 2156422420074752, 54984136073084928, 799573460292407296, 42320889956270080, -425007017470737816576, -15563879892284330213376

Offset: 0

Seiichi Manyama, Feb 20 2025

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

Cf. A205571, A352252, A381280, A381281, A381282, A381283, A381344.

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!*(-2)^((n-k)/2)*a185951(n, k));

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-2)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).

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

A381343 Expansion of e.g.f. exp( sin(sqrt(2)*x) / sqrt(2) ).

Original entry on oeis.org

1, 1, 1, -1, -7, -15, 25, 287, 721, -2847, -30255, -61697, 682761, 5861713, 3105193, -258188513, -1681060063, 4623681473, 135471132705, 564325398271, -6357495670375, -89817656595791, -84337394884167, 7820620314702879, 67277670159083761, -322108989883888479

Offset: 0

Seiichi Manyama, Feb 20 2025

sign

Cf. A002017, A009210, A009229, A351891, A351892, A381277, A381278, A381280, A381344.

Cf. A136630.

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, (-2)^((n-k)/2)*a136630(n, k));

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-2)^k * binomial(n-1,2*k) * a(n-2*k-1).

a(n) = Sum_{k=0..n} (-2)^((n-k)/2) * A136630(n,k).

cp's OEIS Frontend

A381345 Expansion of e.g.f. 1/( 1 - x * cos(sqrt(2)*x) ).

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