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A213554 Principal diagonal of the convolution array A213553.

%I A213554 #15 Sep 08 2022 08:46:02
%S A213554 1,43,334,1406,4271,10577,22764,44220,79437,134167,215578,332410,
%T A213554 495131,716093,1009688,1392504,1883481,2504067,3278374,4233334,
%U A213554 5398855,6807977,8497028,10505780,12877605,15659631,18902898,22662514
%N A213554 Principal diagonal of the convolution array A213553.
%H A213554 G. C. Greubel, <a href="/A213554/b213554.txt">Table of n, a(n) for n = 1..1000</a>
%H A213554 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A213554 a(n) = n*(39*n^4 + 15*n^3 - 25*n^2 + 1)/30.
%F A213554 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
%F A213554 G.f.: x*(1 + 37*x  + 91*x^2 + 27*x^3)/(1 - x)^6.
%t A213554 (* First program *)
%t A213554 b[n_]:= n; c[n_]:= n^3;
%t A213554 T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213554 TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213554 Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t A213554 r[n_]:= Table[T[n, k], {k, 1, 60}]  (* A213553 *)
%t A213554 d = Table[T[n, n], {n, 1, 40}] (* A213554 *)
%t A213554 s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t A213554 s1 = Table[s[n], {n, 1, 50}] (* A101089 *)
%t A213554 (* Second program *)
%t A213554 Table[(39n^5+15n^4-25n^3+n)/30,{n,30}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{1,43,334,1406,4271,10577},30] (* _Harvey P. Dale_, Jan 15 2013 *)
%o A213554 (PARI) vector(30, n, n*(39*n^4 +15*n^3 -25*n^2 +1)/30) \\ _G. C. Greubel_, Jul 31 2019
%o A213554 (Magma) [n*(39*n^4 +15*n^3 -25*n^2 +1)/30: n in [1..30]]; // _G. C. Greubel_, Jul 31 2019
%o A213554 (Sage) [n*(39*n^4 +15*n^3 -25*n^2 +1)/30 for n in (1..30)] # _G. C. Greubel_, Jul 31 2019
%o A213554 (GAP) List([1..30], n-> n*(39*n^4 +15*n^3 -25*n^2 +1)/30); # _G. C. Greubel_, Jul 31 2019
%Y A213554 Cf. A213500, A213553.
%K A213554 nonn,easy
%O A213554 1,2
%A A213554 _Clark Kimberling_, Jun 17 2012