A288867 Numerators of the z-sequence for the Sheffer matrix S2*P = A048993*A007318 = A049020.

1, 0, -1, 3, -53, 65, -1873, 469, -11191, 55391, -1031191, 1334179, -2126212148, 5653970452, -675022695, 438925403269, -67882510220729, 74577815126027, -91314328938731167, 101372762616408631

Offset

0,4

Comments

The denominators are in A288868.

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.

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.

Links

Table of n, a(n) for n=0..19.

Formula

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

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

Example

Recurrence for A049020 from a- and z-sequences:
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; ...
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, ...

Crossrefs

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

Sequence in context: A143387 A228452 A269458 * A082484 A187805 A178608

Adjacent sequences: A288864 A288865 A288866 * A288868 A288869 A288870

Keyword

sign,frac,easy

Author

Wolfdieter Lang, Jun 20 2017

Status

approved