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
Examples
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... ...
Programs
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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)];
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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 *)
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
Extensions
Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010
Comments