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A213583 Principal diagonal of the convolution array A213582.

%I A213583 #32 May 09 2025 23:08:41
%S A213583 1,9,38,120,327,819,1948,4482,10085,22341,48930,106236,229075,491175,
%T A213583 1048184,2227782,4718097,9960921,20970910,44039520,92273951,192937179,
%U A213583 402652308,838859850,1744829437,3623877549,7516191578,15569255172,32212253355,66571991631
%N A213583 Principal diagonal of the convolution array A213582.
%H A213583 Clark Kimberling, <a href="/A213583/b213583.txt">Table of n, a(n) for n = 1..1000</a>
%H A213583 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (7,-19,25,-16,4).
%F A213583 a(n) = 7*a(n-1) - 19*a(n-2) + 25*a(n-3) - 16*a(n-4) + 4*a(n-5).
%F A213583 G.f.: x*(1 + 2*x - 6*x^2) / ((1 - x)^3*(1 - 2*x)^2).
%F A213583 a(n) = (n+1)*(2^(n+2) - 3*n -4)/2. - _Colin Barker_, Nov 04 2017
%F A213583 E.g.f.: (4*(1+2*x)*exp(2*x) - (3*x^2+10*x+4)*exp(x))/2. - _G. C. Greubel_, Jul 08 2019
%t A213583 (* First program *)
%t A213583 b[n_]:= 2^n - 1; c[n_]:= n;
%t A213583 T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213583 TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213583 Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213582 *)
%t A213583 r[n_]:= Table[T[n, k], {k, 40}] (* columns of antidiagonal triangle *)
%t A213583 Table[T[n, n], {n, 1, 40}] (* A213583 *)
%t A213583 s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t A213583 Table[s[n], {n, 1, 50}] (* A156928 *)
%t A213583 (* Second program *)
%t A213583 LinearRecurrence[{7,-19,25,-16,4},{1,9,38,120,327},40] (* _Harvey P. Dale_, Apr 06 2013 *)
%t A213583 Table[(n+1)*(2^(n+2)-3*n-4)/2, {n,40}] (* _G. C. Greubel_, Jul 08 2019 *)
%o A213583 (PARI) Vec(x*(1 + 2*x - 6*x^2) / ((1 - x)^3*(1 - 2*x)^2) + O(x^40)) \\ _Colin Barker_, Nov 04 2017
%o A213583 (PARI) vector(40, n, (n+1)*(2^(n+2) -3*n-4)/2) \\ _G. C. Greubel_, Jul 08 2019
%o A213583 (Magma) [(n+1)*(2^(n+2) -3*n-4)/2: n in [1..40]]; // _G. C. Greubel_, Jul 08 2019
%o A213583 (Sage) [(n+1)*(2^(n+2) -3*n-4)/2 for n in (1..40)] # _G. C. Greubel_, Jul 08 2019
%o A213583 (GAP) List([1..40], n-> (n+1)*(2^(n+2) -3*n-4)/2); # _G. C. Greubel_, Jul 08 2019
%Y A213583 Cf. A213500, A213582.
%K A213583 nonn,easy
%O A213583 1,2
%A A213583 _Clark Kimberling_, Jun 19 2012