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A213759 Principal diagonal of the convolution array A213783.

%I A213759 #29 Aug 27 2018 02:07:44
%S A213759 1,4,11,22,39,62,93,132,181,240,311,394,491,602,729,872,1033,1212,
%T A213759 1411,1630,1871,2134,2421,2732,3069,3432,3823,4242,4691,5170,5681,
%U A213759 6224,6801,7412,8059,8742,9463,10222,11021,11860,12741,13664,14631
%N A213759 Principal diagonal of the convolution array A213783.
%H A213759 Clark Kimberling, <a href="/A213759/b213759.txt">Table of n, a(n) for n = 1..1000</a>
%H A213759 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F A213759 a(n) = (3 - 3*(-1)^n - 4*n + 18*n^2 + 4*n^3)/24.
%F A213759 a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
%F A213759 G.f.:  x*(1 + x + x^2 - x^3)/((1 - x)^4 *(1 + x)).
%F A213759 a(n+1) = a(n) + A047838(n+2) for n > 0. - _Guenther Schrack_, May 24 2018
%F A213759 a(n) = A212964(n+2) - n for n > 0. - _Guenther Schrack_, May 30 2018
%t A213759 b[n_] := Floor[(n + 2)/2]; c[n_] := Floor[(n + 1)/2];
%t A213759 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213759 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213759 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213759 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213783 *)
%t A213759 Table[t[n, n], {n, 1, 40}] (* A213759 *)
%t A213759 LinearRecurrence[{3,-2,-2,3,-1},{1,4,11,22,39},50] (* _Harvey P. Dale_, Jul 22 2014 *)
%Y A213759 Cf. A213783, A213500.
%Y A213759 Partial sums of A047838. - _Guenther Schrack_, May 24 2018
%K A213759 nonn,easy
%O A213759 1,2
%A A213759 _Clark Kimberling_, Jun 22 2012