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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).

%I A143818 #11 Jun 28 2017 20:27:17
%S A143818 0,1,0,-2,-5,-5,20,149,552,991,-3799,-49841,-299937,-1127358,-587744,
%T A143818 34873758,380671819,2584563448,11105613358,-2623056379,-659822835085,
%U A143818 -8393151852216,-69959106516419,-390297675629170,-414406919999723
%N 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).
%C A143818 The coefficients of R(0) and R(2) are listed in A143815 and A143817 respectively.
%F A143818 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
%e A143818 R(n) as a linear combination of R(i),
%e A143818 i = 0..2.
%e A143818 ====================================
%e A143818 ..R(n)..|.....R(0)....R(1)....R(2)..
%e A143818 ====================================
%e A143818 ..R(3)..|.......1......-2.......3...
%e A143818 ..R(4)..|.......6......-5.......7...
%e A143818 ..R(5)..|......25......-5......16...
%e A143818 ..R(6)..|......91......20......46...
%e A143818 ..R(7)..|.....322.....149.....203...
%e A143818 ..R(8)..|....1232.....552....1178...
%e A143818 ..R(9)..|....5672.....991....7242...
%e A143818 ..R(10).|...32202...-3799...43786...
%e A143818 ...
%p A143818 M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
%p A143818 a[0]:=1: b[0]:=0: c[0]:=0:
%p A143818 for n from 1 to M do
%p A143818 a[n]:=add(binomial(n-1,k)*b[k], k=0..n-1);
%p A143818 b[n]:=add(binomial(n-1,k)*c[k], k=0..n-1);
%p A143818 c[n]:=add(binomial(n-1,k)*a[k], k=0..n-1);
%p A143818 end do:
%p A143818 A143818:=[seq(b[n]-c[n], n=0..M)];
%t A143818 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 *)
%Y A143818 A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143816, A143817.
%K A143818 easy,sign
%O A143818 1,4
%A A143818 _Peter Bala_, Sep 03 2008
%E A143818 Spelling/notation corrections by _Charles R Greathouse IV_, Mar 18 2010