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A121868
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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 B sequence (cf. A121867).
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10
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0, -1, -1, 0, 5, 23, 74, 161, -57, -3466, -27361, -155397, -687688, -1888525, 4974059, 134695952, 1400820897, 11055147275, 70658948426, 327448854237, 223871274083, -19116044475298, -314203665206509, -3562429698724513, -33024521386113840, -250403183401213513
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Stirling transform of (I^(n+1)+(-I)^(n+1))/2 = (0,-1,0,1,..) repeated.
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REFERENCES
| A. Fekete and others, Problem 10791, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..250
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
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FORMULA
| This sequence and its companion A121867 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). Furthermore, 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. The precise result is 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, Aug 28 2008]
E.g.f.: A(x) = -sin(exp(x)-1).
a(n) = Sum_{k=0..floor(n/2)} stirling2(n,2*k+1)*(-1)^(k+1). [From Vladimir Kruchinin, Jan 26 2011]
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EXAMPLE
| Contribution from Peter Bala, Aug 28 2008: (Start)
E_2(k) as a 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)
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MAPLE
| # 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)];
## Code based on Stirling transform
stirtr:= proc(p) proc(n) option remember;
add (p(k) *combinat[stirling2](n, k), k=0..n) end
end:
a:= stirtr (n-> (I^(n+1) + (-I)^(n+1))/2):
seq (a(n), n=0..30); # Alois P. Heinz, Jan 29 2011
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CROSSREFS
| Cf. A121867, A024430, A024429.
Cf. A000587, A143623, A143624, A143628, A143631. [From Peter Bala, Aug 28 2008]
Sequence in context: A138905 A125955 A103478 * A111584 A139209 A113946
Adjacent sequences: A121865 A121866 A121867 * A121869 A121870 A121871
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Sep 05 2006
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