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A213575 Antidiagonal sums of the convolution array A213573.

%I A213575 #32 Sep 08 2022 08:46:02
%S A213575 1,10,47,158,441,1098,2539,5590,11909,24818,50967,103662,209521,
%T A213575 421786,846947,1697990,3400893,6807618,13622095,27252190,54513641,
%U A213575 109037930,218088027,436189878,872395381,1744808338,3489636359
%N A213575 Antidiagonal sums of the convolution array A213573.
%H A213575 Clark Kimberling, <a href="/A213575/b213575.txt">Table of n, a(n) for n = 1..500</a>
%H A213575 S. Sykora, <a href="http://dx.doi.org/10.3247/SL1Math06.002">Finite and Infinite Sums of the Power Series (k^p)(x^k)</a>, DOI 10.3247/SL1Math06.002, Section V.
%H A213575 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,16,-9,2).
%F A213575 a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5).
%F A213575 G.f.: x*(1 + 4 x + x^2)/((1 - 2*x)*(1 - x)^4).
%F A213575 From _Stanislav Sykora_, Nov 27 2013: (Start)
%F A213575 a(n) = 2^n*Sum_{k=0..n} k^p*q^k, for p=3, q=1/2.
%F A213575 a(n) = 2^(n+1)*13 - (n^3 + 6*n^2 + 18*n + 26). (End)
%F A213575 a(n) = 2*a(n-1) + n^3. - _Sochima Everton, Biereagu_, Jul 14 2019
%F A213575 E.g.f.: 26*exp(2*x) - (26 +25*x +9*x^2 +x^3)*exp(x). - _G. C. Greubel_, Jul 25 2019
%t A213575 (* First program *)
%t A213575 b[n_]:= 2^(n-1); c[n_]:= n^2;
%t A213575 t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213575 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213575 Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t A213575 r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213573 *)
%t A213575 d = Table[t[n, n], {n, 1, 40}] (* A213574 *)
%t A213575 s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
%t A213575 s1 = Table[s[n], {n, 1, 50}] (* A213575 *)
%t A213575 (* Additional programs *)
%t A213575 Table[Sum[k^3*2^(n-k),{k,0,n}],{n,1,30}] (* _Vaclav Kotesovec_, Nov 28 2013 *)
%o A213575 (PARI) vector(30, n, 13*2^(n+1)-(n^3+6*n^2+18*n+26)) \\ _G. C. Greubel_, Jul 25 2019
%o A213575 (Magma) [13*2^(n+1)-(n^3+6*n^2+18*n+26): n in [1..30]]; // _G. C. Greubel_, Jul 25 2019
%o A213575 (Sage) [13*2^(n+1)-(n^3+6*n^2+18*n+26) for n in (1..30)] # _G. C. Greubel_, Jul 25 2019
%o A213575 (GAP) List([1..30], n-> 13*2^(n+1)-(n^3+6*n^2+18*n+26)); # _G. C. Greubel_, Jul 25 2019
%Y A213575 Cf. A213564, A213500, A232603, A232604.
%K A213575 nonn,easy
%O A213575 1,2
%A A213575 _Clark Kimberling_, Jun 18 2012