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A213574 Principal diagonal of the convolution array A213573.

%I A213574 #25 Sep 08 2022 08:46:02
%S A213574 1,17,93,349,1093,3093,8221,20957,51861,125509,298477,699789,1621285,
%T A213574 3718325,8453181,19069885,42728245,95156901,210762253,464517485,
%U A213574 1019214021,2227173397,4848613213,10519312029,22749902293,49056576773,105495131181,226291086157
%N A213574 Principal diagonal of the convolution array A213573.
%H A213574 Clark Kimberling, <a href="/A213574/b213574.txt">Table of n, a(n) for n = 1..1000</a>
%H A213574 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (9,-33,63,-66,36,-8).
%F A213574 a(n) = 9*a(n-1) - 33*a(n-2) + 63*a(n-3) - 66*a(n-4) + 36*a(n-5) - 8*a(n-6).
%F A213574 G.f.: x*(1 + 8*x - 27*x^2 + 10*x^3 + 16*x^4)/(1 - 3*x + 2*x^2)^3.
%F A213574 a(n) = 2^n*(3+2*n+n^2) - (3+4*n+4*n^2). - _Colin Barker_, Oct 30 2017
%F A213574 E.g.f.: (3+6*x+4*x^2)*exp(2*x) - (3+8*x+4*x^2)*exp(x). - _G. C. Greubel_, Jul 25 2019
%t A213574 (* First program *)
%t A213574 b[n_]:= 2^(n-1); c[n_]:= n;
%t A213574 t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213574 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213574 Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t A213574 r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213568 *)
%t A213574 d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
%t A213574 s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
%t A213574 s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
%t A213574 (* Additional programs *)
%t A213574 LinearRecurrence[{9,-33,63,-66,36,-8},{1,17,93,349,1093,3093},30] (* _Harvey P. Dale_, Jun 25 2014 *)
%t A213574 Rest[CoefficientList[Series[x(1+8x-27x^2+10x^3+16x^4)/(1-3x+2x^2)^3, {x, 0, 30}], x]] (* _Vincenzo Librandi_, Jun 26 2014 *)
%o A213574 (PARI) Vec(x*(1+8*x-27*x^2+10*x^3+16*x^4)/((1-x)^3*(1-2*x)^3) + O(x^30)) \\ _Colin Barker_, Oct 30 2017
%o A213574 (PARI) vector(30, n, 2^n*(3+2*n+n^2) - (3+4*n+4*n^2)) \\ _G. C. Greubel_, Jul 25 2019
%o A213574 (Magma) [2^n*(3+2*n+n^2) - (3+4*n+4*n^2): n in [1..30]]; // _G. C. Greubel_, Jul 25 2019
%o A213574 (Sage) [2^n*(3+2*n+n^2) - (3+4*n+4*n^2) for n in (1..30)] # _G. C. Greubel_, Jul 25 2019
%o A213574 (GAP) List([1..30], n-> 2^n*(3+2*n+n^2) - (3+4*n+4*n^2)); # _G. C. Greubel_, Jul 25 2019
%Y A213574 Cf. A213568, A213500.
%K A213574 nonn,easy
%O A213574 1,2
%A A213574 _Clark Kimberling_, Jun 18 2012