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A163649 Triangle interpolating between (-1)^n (A033999) and A056040(n), read by rows.

%I A163649 #17 Aug 02 2017 05:00:16
%S A163649 1,-1,1,1,-2,2,-1,3,-6,6,1,-4,12,-24,6,-1,5,-20,60,-30,30,1,-6,30,
%T A163649 -120,90,-180,20,-1,7,-42,210,-210,630,-140,140,1,-8,56,-336,420,
%U A163649 -1680,560,-1120,70
%N A163649 Triangle interpolating between (-1)^n (A033999) and A056040(n), read by rows.
%C A163649 Given T(n,k) = (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)!, let A(n,k) = abs(T(n,k)) be the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)*A056040(k)*q^k. Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the extended Motzkin numbers A189912. (See A089627 for the Motzkin numbers and A194586 for the complementary Motzkin numbers.)
%H A163649 G. C. Greubel, <a href="/A163649/b163649.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A163649 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>.
%F A163649 T(n,k) = (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)!.
%F A163649 E.g.f.: egf(x,y) = exp(-x)*BesselI(0,2*x*y)*(1+x*y).
%e A163649 1
%e A163649 -1, 1
%e A163649 1, -2, 2
%e A163649 -1, 3, -6, 6
%e A163649 1, -4, 12, -24, 6
%e A163649 -1, 5, -20, 60, -30, 30
%e A163649 1, -6, 30, -120, 90, -180, 20
%e A163649 -1, 7, -42, 210, -210, 630, -140, 140
%e A163649 1, -8, 56, -336, 420, -1680, 560, -1120, 70
%p A163649 a := proc(n,k) (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)! end:
%p A163649 seq(print(seq(a(n,k),k=0..n)),n=0..8);
%t A163649 t[n_, k_] := (-1)^(n - k)*Floor[k/2]!^(-2)*n!/(n - k)!; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 29 2013 *)
%o A163649 (PARI) for(n=0,10, for(k=0,n, print1((-1)^(n -k)*( (floor(k/2))! )^(-2)*(n!/(n - k)!), ", "))) \\ _G. C. Greubel_, Aug 01 2017
%Y A163649 Row sums give A163650, row sums of absolute values give A163865.
%Y A163649 Aerated versions A194586 (odd case) and A089627 (even case).
%K A163649 sign,tabl
%O A163649 0,5
%A A163649 _Peter Luschny_, Aug 02 2009