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A288869 Numerators of z-sequence for the Sheffer matrix T = P*Lah = A271703 = A007318*A271703 = (exp(t), t/(1-t)). 1

%I

%S 1,-1,4,-21,136,-1045,9276,-93289,1047376,-12975561,175721140,

%T -2581284541,40864292184,-693347907421,12548540320876,

%U -241253367679185,4909234733857696,-105394372192969489,2380337795595885156

%N Numerators of z-sequence for the Sheffer matrix T = P*Lah = A271703 = A007318*A271703 = (exp(t), t/(1-t)).

%C The denominators seem to be the natural numbers A000027.

%C The z-sequence gives the recurrence for column k=0 entries of the triangle T = A271703 (using also lower rows of T): T(n, 0) = Sum_{j=0..n-1} z(j)*T(n-1, j), n >=1, with T(0, 0) = 1. Because column k=0 has e.g.f. exp(x) all entries T(n, 0) = 1, and one obtains rational representations of 1 by the z-sequence recurrence. See the examples.

%F E.g.f. for the rationals r(n): ((1+x)/x)*(1 - exp(-x/(1+x))).

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

%e The rationals r(n) begin: 1, -1/2, 4/3, -21/4, 136/5, -1045/6, 9276/7, -93289/8, 1047376/9, -12975561/10, 175721140/11, -2581284541/12, 40864292184/13, -693347907421/14, 12548540320876/15, ...

%e Recurrence with T= P*Lah = A271703, rational representations of 1:

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

%e 1 = T(3, 0) = 3*(1*1 + (-1/2)*4 + (4/3)*1) = 1

%e 1 = T(4, 0) = 4*(1*1 + (-1/2)*15 + (4/3)*9 + (-21/4)*1) = 1.

%e ...

%Y Cf. A000027, A007318, A271703.

%K sign,easy,frac

%O 0,3

%A _Wolfdieter Lang_, Jun 20 2017

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Last modified September 22 19:36 EDT 2021. Contains 347608 sequences. (Running on oeis4.)