A143818 Let R(n) = sum {k = 0..inf} (3k)^n/(3k)! for n = 0,1,2,... . Then the real number R(n) is an integral linear combination of R(0), R(1) and R(2). This sequence gives the coefficients of R(1).

0, 1, 0, -2, -5, -5, 20, 149, 552, 991, -3799, -49841, -299937, -1127358, -587744, 34873758, 380671819, 2584563448, 11105613358, -2623056379, -659822835085, -8393151852216, -69959106516419, -390297675629170, -414406919999723

Offset

1,4

Comments

The coefficients of R(0) and R(2) are listed in A143815 and A143817 respectively.

Links

Table of n, a(n) for n=1..25.

Formula

a(n) = A143816(n) - A143817(n). a(n) = sum {k = 0..floor((n-1)/3)} (Stirling2(n,3k+1) - Stirling2(n,3k+2)). Let R(n) = sum {k = 0..inf} (3k)^n/(3k)! for n = 0,1,2,... . Then R(n) = A143815(n)*R(0) + A143818(n)*R(1) + A143817(n)*R(2). Some examples are given below. This generalizes the Dobinski relation for the Bell numbers: sum {k = 0..inf} k^n/k! = A000110(n)*exp(1). See A143815 for more details. Compare with A024429, A024430 and A143628--A143631

Example

R(n) as a linear combination of R(i),
i = 0..2.
====================================
..R(n)..|.....R(0)....R(1)....R(2)..
====================================
..R(3)..|.......1......-2.......3...
..R(4)..|.......6......-5.......7...
..R(5)..|......25......-5......16...
..R(6)..|......91......20......46...
..R(7)..|.....322.....149.....203...
..R(8)..|....1232.....552....1178...
..R(9)..|....5672.....991....7242...
..R(10).|...32202...-3799...43786...
...

Maple

M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
a[0]:=1: b[0]:=0: c[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)*c[k], k=0..n-1);
c[n]:=add(binomial(n-1, k)*a[k], k=0..n-1);
end do:
A143818:=[seq(b[n]-c[n], n=0..M)];

Mathematica

m = 24; a[0] = 1; b[0] = 0; c[0] = 0; For[n = 1, n <= m, n++, a[n] = Sum[Binomial[n - 1, k]*b[k], {k, 0, n - 1}]; b[n] = Sum[Binomial[n - 1, k]*c[k], {k, 0, n - 1}]; c[n] = Sum[Binomial[n - 1, k]*a[k], {k, 0, n - 1}] ]; A143818 = Table[c[n] - b[n], {n, 0, m}] (* Jean-François Alcover, Mar 06 2013, after Maple *)

Crossrefs

A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143816, A143817.

Sequence in context: A056396 A085043 A338586 * A238879 A286258 A297446

Adjacent sequences: A143815 A143816 A143817 * A143819 A143820 A143821

Keyword

easy,sign

Author

Peter Bala, Sep 03 2008

Extensions

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

Status

approved