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%I #24 Jul 29 2022 22:26:55
%S 1,9,41,146,456,1312,3568,9312,23552,58112,140544,334336,784384,
%T 1818624,4173824,9494528,21430272,48037888,107020288,237109248,
%U 522715136,1147142144,2507145216,5458886656,11844714496,25618808832,55247372288
%N Third diagonal of triangle A056242.
%H F. K. Hwang and C. L. Mallows, <a href="http://dx.doi.org/10.1016/0097-3165(95)90097-7">Enumerating nested and consecutive partitions</a>, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
%F a(n) = Sum_{0<=j<=n-3} (-1)^(n-3-j)*binomial(n-3, j)*binomial(n+2j-1, 2j), for n>=3. - Pab Ter (pabrlos2(AT)yahoo.com), Nov 06 2005
%F Conjecture: a(n) = 2^(-6+n)*(32-35*n+9*n^2). G.f.: x^3*(1+3*x-x^2)/(1-2*x)^3. - _Colin Barker_, Mar 20 2012
%p seq(add((-1)^(n-3-j)*binomial(n-3,j)*binomial(n+2*j-1,2*j),j=0..n-3),n=3..40); # Pab Ter (pabrlos2(AT)yahoo.com), Nov 06 2005
%p T:=proc(n,k) local j: if k=1 then 1 elif k<=n then add((-1)^(k-1-j)*binomial(k-1,j)*binomial(n+2*j-1,2*j),j=0..k-1) else 0 fi end: seq(T(n,n-2),n=3..40); # Pab Ter (pabrlos2(AT)yahoo.com), Nov 06 2005
%Y Cf. A056242.
%K nonn,easy
%O 3,2
%A _Colin Mallows_, Aug 23 2000
%E More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 06 2005
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