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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). 13
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 (list; graph; refs; listen; history; text; internal format)
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

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.. 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. - Peter Bala, Aug 28 2008

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

Contribution 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-Fran├žois Alcover, Mar 06 2013, after Alois P. Heinz *)

CROSSREFS

Cf. A121868, A024430, A024429.

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

Sequence in context: A247569 A115389 A303564 * A300673 A009193 A144253

Adjacent sequences:  A121864 A121865 A121866 * A121868 A121869 A121870

KEYWORD

sign

AUTHOR

N. J. A. Sloane, Sep 05 2006

STATUS

approved

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Last modified May 26 05:25 EDT 2019. Contains 323579 sequences. (Running on oeis4.)