A090145 Even-indexed terms of the binomial transform equal 1 and the odd-indexed terms of the second binomial transform equal 1.

1, -1, 2, -7, 20, -31, 2, -127, 1640, -511, -49498, -2047, 2706860, -8191, -199344598, -32767, 19391577680, -131071, -2404879413298, -524287, 370371189286100, -2097151, -69348874388943598, -8388607, 15514534163573864120, -33554431, -4087072509293056783498

Offset: 0

Author

Paul D. Hanna, Nov 22 2003

Keywords

• sign

Examples

			a(11) = 1 - 2^11 = -2047.
First binomial  = {1, 0, 1, -3, 1, 15, 1, -273, 1, 7935, 1, -353793,..};
Second binomial = {1, 1, 2, 1, -4, 1, 62, 1, -1384, 1, 50522, 1, -2702764,..}.
		

Crossrefs

Cf. A090158.

Formula

E.g.f.: 2*((1 + sinh(x))*exp(x) - sinh(x)) / (exp(x)*(1 + exp(2*x))).

a(2n+1) = 1 - 2^(2n+1);

1 = sum_{k=0..2n} C(2n, k)*a(k);

1 = sum_{k=0..2n-1} 2^(2n-1-k)*C(2n-1, k)*a(k).