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A213581 Antidiagonal sums of the convolution array A213571.

%I A213581 #26 Sep 08 2022 08:46:02
%S A213581 1,8,36,124,367,988,2498,6048,14197,32576,73472,163508,360027,785908,
%T A213581 1703294,3669240,7863393,16776120,35650300,75495980,159381831,
%U A213581 335542348,704640826,1476392464,3087004877,6442447728,13421769208
%N A213581 Antidiagonal sums of the convolution array A213571.
%H A213581 Clark Kimberling, <a href="/A213581/b213581.txt">Table of n, a(n) for n = 1..500</a>
%H A213581 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,-26,44,-41,20,-4).
%F A213581 a(n) = 8*a(n-1) - 26*a(n-2) + 44*a(n-3) - 41*a(n-4) + 20*a(n-5) - 4*a(n-6).
%F A213581 G.f.: f(x)/g(x), where f(x) = x*(1 - 2*x^2) and g(x) = (1 - x)^4*(1 - 2*x)^2.
%F A213581 a(n) = 8 +(n-2)*2^(n+2) -(n-2)*n*(n+5)/6. - _Bruno Berselli_, Jul 09 2012
%t A213581 (* First Program *)
%t A213581 b[n_]:= n; c[n_]:= -1 + 2^n;
%t A213581 t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213581 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213581 Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t A213581 r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213571 *)
%t A213581 d = Table[t[n, n], {n, 1, 40}] (* A213572 *)
%t A213581 s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
%t A213581 s1 = Table[s[n], {n, 1, 50}] (* A213581 *)
%t A213581 (* Second program *)
%t A213581 Table[2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6, {n,35}] (* _G. C. Greubel_, Jul 26 2019 *)
%o A213581 (PARI) vector(35, n, 2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6) \\ _G. C. Greubel_, Jul 26 2019
%o A213581 (Magma) [2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6: n in [1..35]]; // _G. C. Greubel_, Jul 26 2019
%o A213581 (Sage) [2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6 for n in (1..35)] # _G. C. Greubel_, Jul 26 2019
%o A213581 (GAP) List([1..35], n-> 2^(n+2)*(n-2) - (n^3+3*n^2-10*n-48)/6); # _G. C. Greubel_, Jul 26 2019
%Y A213581 Cf. A213571, A213500.
%K A213581 nonn,easy
%O A213581 1,2
%A A213581 _Clark Kimberling_, Jun 19 2012