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A047922 Triangle of numbers a(n,k) = number of terms in n X n determinant with 2 adjacent diagonals of k and k-1 0's (0<=k<=n).

%I A047922 #23 Mar 28 2018 18:03:33
%S A047922 1,1,0,2,1,0,6,4,1,1,24,18,8,5,3,120,96,54,34,23,16,720,600,384,258,
%T A047922 182,131,96,5040,4320,3000,2136,1566,1168,883,675,40320,35280,25920,
%U A047922 19320,14664,11274,8756,6859,5413,362880,322560,246960,190800,149160,117696,93582,74902,60301,48800
%N A047922 Triangle of numbers a(n,k) = number of terms in n X n determinant with 2 adjacent diagonals of k and k-1 0's (0<=k<=n).
%H A047922 Alois P. Heinz, <a href="/A047922/b047922.txt">Rows n = 0..140, flattened</a>
%H A047922 J. D. H. Dickson, <a href="http://plms.oxfordjournals.org/content/s1-10/1/120.extract">Discussion of two double series arising from the number of terms in determinants of certain forms</a>, Proc. London Math. Soc., 10 (1879), 120-122.
%H A047922 J. D. H. Dickson, <a href="/A002775/a002775.pdf">Discussion of two double series arising from the number of terms in determinants of certain forms</a>, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]
%F A047922 Right diagonal is A000271, column k=0 is A000142; other entries given by a(n, k) = a(n, k+1) + 2a(n-1, k) + a(n-2, k-1).
%e A047922 Triangle starts:
%e A047922   1;
%e A047922   1, 0;
%e A047922   2, 1, 0;
%e A047922   6, 4, 1, 1;
%e A047922   ...
%p A047922 a:= proc(n, k) option remember; `if`(k=0, n!, `if`(n=k,
%p A047922       `if`(n<3, (n-1)*(n-2)/2, (n-1)*(a(n-1$2)+a(n-2$2))
%p A047922       +a(n-3$2)), a(n, k+1) +2*a(n-1, k) +a(n-2, k-1)))
%p A047922     end:
%p A047922 seq(seq(a(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Jun 24 2017
%t A047922 a[n_, n_] := (-1)^n*HypergeometricPFQ[{1, -n, n+1}, {1/2}, 1/4]; a[n_, k_] := a[n, k] = a[n, k+1] + 2*a[n-1, k] + a[n-2, k-1]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2015 *)
%Y A047922 Columns give A000142, A001563, A002775, A002776. Cf. A047920.
%K A047922 nonn,tabl,nice,easy
%O A047922 0,4
%A A047922 _N. J. A. Sloane_
%E A047922 More terms from Larry Reeves (larryr(AT)acm.org), Sep 29 2000