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%I #41 Oct 15 2023 02:59:50
%S 1,0,-1,-3,-6,-5,33,266,1309,4905,11516,-22935,-556875,-4932512,
%T -32889885,-174282151,-612400262,907955295,45283256165,573855673458,
%U 5397236838345,41604258561397,250231901787780,756793798761989,-8425656230853383,-213091420659985440,-2990113204010882473
%N Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)*A(k); entry gives A sequence (cf. A121868).
%C Stirling transform of A056594.
%H Alois P. Heinz, <a href="/A121867/b121867.txt">Table of n, a(n) for n = 0..220</a>
%H A. Fekete and G. Martin, <a href="http://www.jstor.org/stable/2695545">Problem 10791: Squared Series Yielding Integers</a>, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.
%F From _Peter Bala_, Aug 28 2008: (Start)
%F 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 generalizations of the Uppuluri-Carpenter numbers (complementary Bell numbers) A000587. Define E_2(k) = Sum_{n >= 0} (-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.
%F To find the precise result, show F(k) := Sum_{n >= 0} (-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. (End)
%F E.g.f.: A(x) = cos(exp(x)-1).
%F a(n) = Sum_{k=0..floor(n/2)} stirling2(n,2*k)*(-1)^(k). - _Vladimir Kruchinin_, Jan 29 2011
%e From _Peter Bala_, Aug 28 2008: (Start)
%e E_2(k) as linear combination of E_2(i), i = 0..1.
%e ============================
%e ..E_2(k)..|...E_2(0)..E_2(1)
%e ============================
%e ..E_2(2)..|....-1.......1...
%e ..E_2(3)..|....-3.......0...
%e ..E_2(4)..|....-6......-5...
%e ..E_2(5)..|....-5.....-23...
%e ..E_2(6)..|....33.....-74...
%e ..E_2(7)..|...266....-161...
%e ..E_2(8)..|..1309......57...
%e ..E_2(9)..|..4905....3466...
%e ...
%e (End)
%p # Maple code for A024430, A024429, A121867, A121868.
%p 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;
%p 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)];
%p # Code based on Stirling transform:
%p stirtr:= proc(p) proc(n) option remember;
%p add(p(k) *Stirling2(n,k), k=0..n) end
%p end:
%p a:= stirtr(n-> (I^n + (-I)^n)/2):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jan 29 2011
%t a[n_] := (BellB[n, -I] + BellB[n, I])/2; Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Mar 06 2013, after _Alois P. Heinz_ *)
%o (PARI) a(n) = sum(k=0,n\2, (-1)^k*stirling(n,2*k,2));
%o vector(30, n, a(n-1)) \\ _G. C. Greubel_, Oct 09 2019
%o (Magma) [(&+[(-1)^k*StirlingSecond(n,2*k): k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Oct 09 2019
%o (Sage) [sum((-1)^k*stirling_number2(n,2*k) for k in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, Oct 09 2019
%o (GAP) List([0..30], n-> Sum([0..Int(n/2)], k-> (-1)^k*Stirling2(n,2*k)) ); # _G. C. Greubel_, Oct 09 2019
%Y Cf. A121868, A024430, A024429, A056594.
%Y Cf. A000587, A143623, A143624, A143628, A143631. - _Peter Bala_, Aug 28 2008
%K sign,easy
%O 0,4
%A _N. J. A. Sloane_, Sep 05 2006
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