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