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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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14
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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
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OFFSET
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0,5
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COMMENTS
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Stirling transform of (I^(n+1)+(-I)^(n+1))/2 = (0,-1,0,1,..) repeated.
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LINKS
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FORMULA
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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 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). 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. (End)
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). (End)
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EXAMPLE
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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
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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) *Stirling2(n, k), k=0..n) end
end:
a:= stirtr(n-> (I^(n+1) + (-I)^(n+1))/2):
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MATHEMATICA
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stirtr[p_] := Module[{f}, f[n_] := f[n] = Sum[p[k]*StirlingS2[n, k], {k, 0, n}]; f]; a = stirtr[(I^(#+1)+(-I)^(#+1))/2&]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)
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PROG
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(PARI) a(n) = sum(k=0, n\2, (-1)^(k+1)*stirling(n, 2*k+1, 2));
(Magma) [(&+[(-1)^(k+1)*StirlingSecond(n, 2*k+1): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Oct 09 2019
(Sage) [sum((-1)^(k+1)*stirling_number2(n, 2*k+1) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 09 2019
(GAP) List([0..30], n-> Sum([0..Int(n/2)], k-> (-1)^(k+1)* Stirling2(n, 2*k+1)) ); # G. C. Greubel, Oct 09 2019
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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