0,4

A. Fekete and others, Problem 10791, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.

This sequence and its companion A121868 are related to the pair of constants cos(1) + sin(1) and cos(1) - sin(1) and may be viewed as generalisations of the Uppuluri-Carpenter numbers (complementary Bell numbers) A000587. Define E_2(k) = sum {n = 0.. inf} (-1)^floor(n/2)*n^k/n! for k = 0,1,2,... . Then E_2(0) = cos(1) + sin(1) and E_2(1) = cos(1) - sin(1). It is easy to see that E_2(k+2) = E_2(k+1) - sum {i = 0..k} 2^i*binomial(k,i)*E_2(k-i) for k >= 0. Hence E_2(k) is an integral linear combination of E_2(0) and E_2(1) (a Dobinski-type relation). For example, E_2(2) = - E_2(0) + E_2(1), E_2(3) = -3*E_2(0) and E_2(4) = - 6*E_2(0) - 5*E_2(1). More examples are given below.

To find the precise result, show F(k) := sum {n = 0.. inf} (-1)^floor((n+1)/2)*n^k/n! satisfies the above recurrence with F(0) = E_2(1) and F(1) = -E_2(0) and then use the identity sum {i = 0..k} binomial(k,i)*E_2(i) = -F(k+1) to obtain E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1). For similar results see A143628. The decimal expansions of E_2(0) and E_2(1) are given in A143623 and A143624 respectively. [From Peter Bala (pbala(AT)toucansurf.com), Aug 28 2008]

Contribution from Peter Bala (pbala(AT)toucansurf.com), Aug 28 2008: (Start)

E_2(k) as linear combination of E_2(i), i = 0..1.

============================

..E_2(k)..|...E_2(0)..E_2(1)

============================

..E_2(2)..|....-1.......1...

..E_2(3)..|....-3.......0...

..E_2(4)..|....-6......-5...

..E_2(5)..|....-5.....-23...

..E_2(6)..|....33.....-74...

..E_2(7)..|...266....-161...

..E_2(8)..|..1309......57...

..E_2(9)..|..4905....3466...

...

(End)

# Maple code for A024430, A024429, A121867, A121868.

M:=30; a:=array(0..100); b:=array(0..100); c:=array(0..100); d:=array(0..100); a[0]:=1; b[0]:=0; c[0]:=1; d[0]:=0;

for n from 1 to M do a[n]:=add(binomial(n-1, k)*b[k], k=0..n-1); b[n]:=add(binomial(n-1, k)*a[k], k=0..n-1); c[n]:=add(binomial(n-1, k)*d[k], k=0..n-1); d[n]:=-add(binomial(n-1, k)*c[k], k=0..n-1); od: ta:=[seq(a[n], n=0..M)]; tb:=[seq(b[n], n=0..M)]; tc:=[seq(c[n], n=0..M)]; td:=[seq(d[n], n=0..M)];

Cf. A121868, A024430, A024429.

A000587, A143623, A143624, A143628, A143631. [From Peter Bala (pbala(AT)toucansurf.com), Aug 28 2008]

Sequence in context: A095359 A048724 A115389 this_sequence A009193 A144253 A138743

Adjacent sequences: A121864 A121865 A121866 this_sequence A121868 A121869 A121870

sign

N. J. A. Sloane (njas(AT)research.att.com), Sep 05 2006