A051055 - OEIS

%I #26 Aug 03 2017 02:53:15

%S 0,1,0,1,2,59,1092,51412,3420384,382912420,68021283668,19474443244283,

%T 9025228384142396,6825775070789988992,8486240219059861120000,

%U 17454179683586670023001218,59698062960218238908531091872

%N 'Connected' alternating sign n X n matrices, i.e., not made from smaller blocks.

%C A003827 factors out the singleton components only, but many alternating sign matrices can be decomposed into larger pieces.

%H Alois P. Heinz, <a href="/A051055/b051055.txt">Table of n, a(n) for n = 0..90</a>

%F Sum_{k>=0} a(k)z^k/k!^2 = log(Sum_{k>=0} r(k)z^k/k!^2) where r(k) is the k-th Robbins number A005130(n).

%F a(n) = r(n) - (1/n)*Sum_{k=0..n-1} k*binomial(n, k)^2*r(n-k)*a(k), n > 0, a(0)=0, where r(k) is the k-th Robbins number A005130(n). - _Vladeta Jovovic_, Mar 16 2000

%e a(4)=2 because of the alternating sign matrices {{0,1,0,0},{1,-1,1,0},{0,1,-1,1},{0,0,1,0}} and {{0,0,1,0},{0,1,-1,1},{1,-1,1,0},{0,1,0,0}}.

%t r[n_] = Product[(3k+1)!/(n+k)!, {k, 0, n-1}] ; a[n_] := a[n] = r[n] - (1/n)*Sum[k*Binomial[n, k]^2*r[n-k]*a[k], {k, 0, n-1}]; a[0] = 0; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Aug 01 2011, after _Vladeta Jovovic_ *)

%Y Cf. A003827, A005130.

%K nice,easy,nonn

%O 0,5

%A _Don Knuth_

%E More terms from _Vladeta Jovovic_, Mar 16 2000