A121867 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).

1, 0, -1, -3, -6, -5, 33, 266, 1309, 4905, 11516, -22935, -556875, -4932512, -32889885, -174282151, -612400262, 907955295, 45283256165, 573855673458, 5397236838345, 41604258561397, 250231901787780, 756793798761989, -8425656230853383, -213091420659985440, -2990113204010882473

Offset

0,4

Comments

Stirling transform of A056594.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..220

A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.

Formula

From Peter Bala, Aug 28 2008: (Start)

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.

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)

E.g.f.: A(x) = cos(exp(x)-1).

a(n) = Sum_{k=0..floor(n/2)} stirling2(n,2*k)*(-1)^(k). - Vladimir Kruchinin, Jan 29 2011

Example

From Peter Bala, 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

# 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) *Stirling2(n, k), k=0..n) end
         end:
a:= stirtr(n-> (I^n + (-I)^n)/2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 29 2011

Mathematica

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 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (-1)^k*stirling(n, 2*k, 2));
vector(30, n, a(n-1)) \\ G. C. Greubel, Oct 09 2019
(Magma) [(&+[(-1)^k*StirlingSecond(n, 2*k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Oct 09 2019
(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
(GAP) List([0..30], n-> Sum([0..Int(n/2)], k-> (-1)^k*Stirling2(n, 2*k)) ); # G. C. Greubel, Oct 09 2019

Crossrefs

Cf. A121868, A024430, A024429, A056594.

Cf. A000587, A143623, A143624, A143628, A143631. - Peter Bala, Aug 28 2008

Sequence in context: A247569 A115389 A303564 * A307132 A300673 A335633

Adjacent sequences: A121864 A121865 A121866 * A121868 A121869 A121870

Keyword

sign,easy

Author

N. J. A. Sloane, Sep 05 2006

Status

approved