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A144381 a(n) = A142458(n+5, n). Fifth diagonal of A142458 triangle.

%I A144381 #7 Mar 16 2022 02:55:27
%S A144381 1,677,47175,1709675,44451550,947113254,17716715490,302925749370,
%T A144381 4856552119935,74258231957275,1095758678253041,15736592058221517,
%U A144381 221321453958111620,3062416225698505060,41836761536767296660,565817483249269872324,7591501608353930033805,101209790951020335444705
%N A144381 a(n) = A142458(n+5, n). Fifth diagonal of A142458 triangle.
%H A144381 G. C. Greubel, <a href="/A144381/b144381.txt">Table of n, a(n) for n = 1..890</a>
%H A144381 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (75, -2520, 50204, -661458, 6086718, -40273648, 194372208, -687083013, 1771618303, -3293261472, 4325310828, -3886563008, 2261691264, -765434880, 114150400).
%F A144381 a(n) = A142458(n+5, n).
%F A144381 From _G. C. Greubel_, Mar 16 2022: (Start)
%F A144381 G.f.: x*(1 +602*x -1080*x^2 -172614*x^3 +1780275*x^4 -5025348*x^5 -7549548*x^6 +60043488*x^7 -99645984*x^8 +39979520*x^9 +27596800*x^10)/((1-x)^5*(1-4*x)^4*(1-7*x)^3*(1-10*x)^2*(1-13*x)).
%F A144381 a(n) = (1/1944)*((27*n^3 +216*n^2 +549*n +440)*(3*n +2 - 2*4^(n+4)) +
%F A144381    60*(9*n^2 +57*n +88)*7^(n+3) -32*(3*n+11)*10^(n+4) + 880*13^(n+3)). (End)
%t A144381 T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k-m+1)*T[n-1, k, m]];
%t A144381 A144381[n_]:= T[n+4,n,3];
%t A144381 Table[A144381[n], {n,30}] (* modified by _G. C. Greubel_, Mar 16 2022 *)
%o A144381 (Magma) [(1/1944)*((27*n^3 + 216*n^2 + 549*n + 440)*(3*n + 2 - 2*4^(n + 4)) +
%o A144381    60*(9*n^2 + 57*n + 88)*7^(n + 3) - 32*(3*n + 11)*10^(n + 4) +
%o A144381 880*13^(n + 3)): n in [1..30]]; // _G. C. Greubel_, Mar 16 2022
%o A144381 (Sage)
%o A144381 @CachedFunction
%o A144381 def T(n,k,m): # A144381
%o A144381     if (k==1 or k==n): return 1
%o A144381     else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
%o A144381 def A144381(n): return T(n+4, n, 3)
%o A144381 [A144381(n) for n in (1..30)] # _G. C. Greubel_, Mar 16 2022
%Y A144381 Cf. A142458, A142976, A144380, A144414.
%K A144381 nonn
%O A144381 1,2
%A A144381 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 01 2008
%E A144381 Edited by _G. C. Greubel_, Mar 16 2022