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%I #13 Jul 13 2022 17:57:53
%S 1,2,3,6,14,37,105,312,956,2996,9554,30897,101083,333947,1112497,
%T 3732956,12605030,42800318,146046820,500555448,1722402304,5948047170,
%U 20607691518,71610355541,249520257107,871614139397,3051737703527
%N Transform of Catalan numbers whose Hankel transform satisfies the Somos-4 recurrence.
%C Hankel transform is A157101.
%C The ratio of this generating function by the generating function of A025262 is x*(1-x), which means this sequence is the partial sums of A025262. - _Sean A. Irvine_, _R. J. Mathar_, Jun 27 2022
%H G. C. Greubel, <a href="/A157100/b157100.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (1+x)*c(x*(1-x^2)), c(x) the g.f. of A000108;
%F a(n) = Sum_{k=0..n} (-1)^binomial(n-k,2)*binomial(k,floor((n-k)/2))*A000108(k).
%F Conjecture: (n+1)*a(n) +(-5*n+1)*a(n-1) +2*(2*n-1)*a(n-2) +2*(2*n-7)*a(n-3) +2*(-2*n+7)*a(n-4) = 0. - _R. J. Mathar_, Feb 05 2015
%t a[n_]:= Sum[(-1)^Binomial[k, 2]*Binomial[n-k, Floor[k/2]]*CatalanNumber[n-k], {k,0,n}];
%t Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, Jan 11 2022 *)
%o (Sage)
%o def A157100(n): return sum((-1)^binomial(k,2)*binomial(n-k, k//2)*catalan_number(n-k) for k in (0..n))
%o [A157100(n) for n in (0..40)] # _G. C. Greubel_, Jan 11 2022
%Y Cf. A000108, A157101.
%Y Partial sums of A025262.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Feb 22 2009
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