A288867 - OEIS

%I #13 Jun 21 2017 11:58:55

%S 1,0,-1,3,-53,65,-1873,469,-11191,55391,-1031191,1334179,-2126212148,

%T 5653970452,-675022695,438925403269,-67882510220729,74577815126027,

%U -91314328938731167,101372762616408631

%N Numerators of the z-sequence for the Sheffer matrix S2*P = A048993*A007318 = A049020.

%C The denominators are in A288868.

%C For a- and z-sequences for Sheffer matrices (infinite lower triangular) see a W. Lang link under A006232, also for references. The a-sequence for Sheffer (exp(exp(x)-1), exp(x) - 1), given in A049020 is a = A006232/A006233.

%C The combined recurrence for A049020 from these a- and z-sequences is: T(n, 0) = n*Sum_{j=0..n-1} z(j)*T(n-1, j), n >= 1,  T(0, 0) = 1; T(n, k) = 0 if k < n, T(n, k) = (n/k)*Sum_{j=0..n-k} a(n)*T(n-1, k-1+j), n >= k >= 1.

%F E.g.f. of r(n) = a(n)/A288868(n), n >= 0: (exp(x)-1)/(log(1+x)*exp(x)).

%F a(n) = numerator(r(n)) (r(n) in lowest terms).

%e Recurrence for A049020 from a- and z-sequences:

%e T(1, 0) = 1*1*1 =1; T(1, 1) = (1/1)*1*1*1 = 1, T(2, 0) = 2*(1*1 + 0*1) = 2,  T(2, 1) = (2/1)*(1*1*1 + 1*(1/2)*1) = 3, T(2, 2) = (2/2)*1**1 = 1; ...

%e The rationals r(n) begin: 1, 0, -1/3, 3/4, -53/30, 65/12, -1873/84, 469/4, -11191/15, 55391/10, -1031191/22, 1334179/3, -2126212148/455, ...

%Y Cf. A006232/A006233, A007318, A048993, A049020, A288868.

%K sign,frac,easy

%O 0,4

%A _Wolfdieter Lang_, Jun 20 2017