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A070543 Triangular array read by rows: T(n,k) = number of k-dimensional isotropic subspaces of Spin(2n+1,C), n >= 1, 1 <= k <= n.

%I A070543 #29 Feb 16 2022 23:21:56
%S A070543 1,3,3,5,7,6,7,11,12,10,9,15,18,18,15,11,19,24,26,25,21,13,23,30,34,
%T A070543 35,33,28,15,27,36,42,45,45,42,36,17,31,42,50,55,57,56,52,45,19,35,48,
%U A070543 58,65,69,70,68,63,55,21,39,54,66,75,81,84,84,81,75,66,23,43,60,74,85,93
%N A070543 Triangular array read by rows: T(n,k) = number of k-dimensional isotropic subspaces of Spin(2n+1,C), n >= 1, 1 <= k <= n.
%H A070543 G. C. Greubel, <a href="/A070543/b070543.txt">Rows n = 1..100 of triangle, flattened</a>
%H A070543 John Baez, <a href="http://math.ucr.edu/home/baez/week181.html">Week 181</a>
%F A070543 T(n, k) = k*(k+1)/2 + 2*k*(n-k) if 0 < k <= n.
%F A070543 G.f.: (1+x-2*x^2*y)/((1-x)^2*(1-x*y)^3). - _Vladeta Jovovic_, Mar 05 2004
%F A070543 T(n, k) = A141419(2*n-k, k). - _Peter Munn_, Aug 21 2019
%e A070543 Rows:
%e A070543    1;
%e A070543    3,  3;
%e A070543    5,  7,  6;
%e A070543    7, 11, 12, 10;
%e A070543    9, 15, 18, 18, 15;
%e A070543   11, 19, 24, 26, 25, 21;
%e A070543   ...
%p A070543 T:=(n,k) -> k*(k+1)/2+2*k*(n-k); r:=n->[seq(T(n,k),k=1..n)]; for r from 1 to 12 do lprint(r(n)); od: # _N. J. A. Sloane_, Aug 21 2019
%t A070543 nmax = 12; t[n_, k_] := If[k < 1 || k > n, 0, k*(k+1)/2 + 2*k*(n-k)]; Flatten[ Table[t[n , k], {n, 1, nmax}, {k, 1, n}]] (* _Jean-François Alcover_, Oct 19 2011, after PARI *)
%o A070543 (PARI) {T(n, k) = if( k<1 || k>n, 0, k * (k + 1) / 2 + 2 * k * (n - k))}
%o A070543 (Magma) [k*(k+1 + 4*(n-k))/2: k in [1..n], n in [1..12]]; // _G. C. Greubel_, Sep 05 2019
%o A070543 (Sage) [[k*(k+1 + 4*(n-k))/2 for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Sep 05 2019
%o A070543 (GAP) Flat(List([1..12], n-> List([1..n], k-> k*(k+1 + 4*(n-k))/2 ))); # _G. C. Greubel_, Sep 05 2019
%Y A070543 Cf. A141419.
%K A070543 nonn,tabl,easy,nice
%O A070543 1,2
%A A070543 _Michael Somos_, Apr 28 2002
%E A070543 Offset changed to 1 by _N. J. A. Sloane_, Aug 21 2019