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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
1, -1, 4, -21, 136, -1045, 9276, -93289, 1047376, -12975561, 175721140, -2581284541, 40864292184, -693347907421, 12548540320876, -241253367679185, 4909234733857696, -105394372192969489, 2380337795595885156 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The denominators seem to be the natural numbers A000027.

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.

LINKS

Table of n, a(n) for n=0..18.

FORMULA

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

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

EXAMPLE

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, ...

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

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

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

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

...

CROSSREFS

Cf. A000027, A007318, A271703.

Sequence in context: A292928 A209881 A052852 * A288268 A265952 A121124

Adjacent sequences:  A288866 A288867 A288868 * A288870 A288871 A288872

KEYWORD

sign,easy,frac

AUTHOR

Wolfdieter Lang, Jun 20 2017

STATUS

approved

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Last modified September 22 16:59 EDT 2019. Contains 327311 sequences. (Running on oeis4.)