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A213572 Principal diagonal of the convolution array A213571.

%I A213572 #16 Sep 08 2022 08:46:02
%S A213572 1,13,82,406,1809,7659,31588,128476,518611,2084809,8361918,33497010,
%T A213572 134094757,536608663,2146926472,8588754808,34357247847,137433710421,
%U A213572 549744803650,2199000186670,8796044787481,35184271425283
%N A213572 Principal diagonal of the convolution array A213571.
%H A213572 Clark Kimberling, <a href="/A213572/b213572.txt">Table of n, a(n) for n = 1..1000</a>
%H A213572 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (11,-47,101,-116,68,-16).
%F A213572 a(n) = (2^(n+2)*(2^n-1) - (2^(n+1) + n + 1)*n)/2.
%F A213572 a(n) = 11*a(n-1) - 47*a(n-2) + 101*a(n-3) - 116*a(n-4) + 68*a(n-5) - 16*a(n-6).
%F A213572 G.f.:  f(x)/g(x), where f(x) = x*(1 + 2*x - 14*x^2 + 14*x^3) and g(x) = (1 - 4*x)*((1 - x)^3)*(1 - 2*x)^2.
%t A213572 (* First program *)
%t A213572 b[n_]:= n; c[n_]:= -1 + 2^n;
%t A213572 t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213572 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213572 Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t A213572 r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213571 *)
%t A213572 d = Table[t[n, n], {n, 1, 40}] (* A213572 *)
%t A213572 s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
%t A213572 s1 = Table[s[n], {n, 1, 50}] (* A213581 *)
%t A213572 (* Additional programs *)
%t A213572 Table[2^(2*n+1) -2^n*(n+2)-Binomial[n+1, 2], {n,30}] (* _G. C. Greubel_, Jul 25 2019 *)
%o A213572 (PARI) vector(30, n, 2^(2*n+1) -2^n*(n+2) -binomial(n+1, 2)) \\ _G. C. Greubel_, Jul 25 2019
%o A213572 (Magma) [2^(2*n+1) -2^n*(n+2) -Binomial(n+1, 2): n in [1..30]]; // _G. C. Greubel_, Jul 25 2019
%o A213572 (Sage) [2^(2*n+1) -2^n*(n+2) -binomial(n+1, 2) for n in (1..30)] # _G. C. Greubel_, Jul 25 2019
%o A213572 (GAP) List([1..30], n-> 2^(2*n+1) -2^n*(n+2) -Binomial(n+1, 2)); # _G. C. Greubel_, Jul 25 2019
%Y A213572 Cf. A213571, A213500.
%K A213572 nonn,easy
%O A213572 1,2
%A A213572 _Clark Kimberling_, Jun 19 2012