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%I #12 Nov 13 2012 05:55:28
%S 1,1,-2,-19,-74,-68,1856,22717,182806,1095506,2706452,-62235754,
%T -1630900556,-28213310474,-422792067164,-5747245586467,
%U -68720160772442,-602550199498622,1056275553274100,251539588303778798,9237652624016037908,263685036472764512992
%N Central terms in rows of triangle A219120.
%C The number of contiguous signs of the terms seems to grow roughly in proportion to the square-root of the number of terms.
%H Paul D. Hanna, <a href="/A219121/b219121.txt">Table of n, a(n) for n = 1..200</a>
%F a(n) = [x^n] (1-x)^(2*n-1) * Sum_{k>=0} k^n *(k+1)^(k-1) * exp(-(k+1)*x) * x^k/k!.
%e Triangle A219120 begins:
%e 1;
%e 1, 1, -1;
%e 1, 5, -2, -2, 1;
%e 1, 15, 13, -19, 3, 3, -1;
%e 1, 37, 128, -26, -74, 46, -4, -4, 1;
%e 1, 83, 679, 755, -654, -68, 230, -90, 5, 5, -1;
%e 1, 177, 2866, 9048, 2091, -5741, 1856, 498, -545, 155, -6, -6, 1; ...
%e in which the o.g.f. of row n, R(x,n), is given by:
%e R(x,n) = (1-x)^(2*n-1) * Sum_{k>=0} k^n *(k+1)^(k-1) * exp(-(k+1)*x) * x^k/k!;
%e note that the coefficient of x^n in R(x,n), for n>=1, forms this sequence.
%e The signs of the terms of this sequence begin:
%e +,+,
%e -,-,-,-,
%e +,+,+,+,+,
%e -,-,-,-,-,-,-,
%e +,+,+,+,+,+,+,+,+,+,
%e -,-,-,-,-,-,-,-,-,-,-,
%e +,+,+,+,+,+,+,+,+,+,+,+,+,
%e -,-,-,-,-,-,-,-,-,-,-,-,-,-,
%e +,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,
%e -,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,
%e +,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,
%e -,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,
%e +,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+, ...
%o (PARI) {a(n)=polcoeff((1-x)^(2*n-1)*sum(k=0,2*n,(k^n)*(k+1)^(k-1)*x^k/k!*exp(-(k+1)*x +x*O(x^n))),n)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A219120.
%K sign
%O 1,3
%A _Paul D. Hanna_, Nov 13 2012
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