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A386956 a(n) = Sum_{k=0..n} (k+1) * 8^k * binomial(2*n+1,n-k).

%I A386956 #15 Sep 03 2025 09:10:25
%S A386956 1,19,282,3763,47294,571950,6733668,77723187,883589238,9924844474,
%T A386956 110396411372,1218075749934,13348677037868,145438914042172,
%U A386956 1576690043132376,17018212213758771,182983432175308710,1960781840268630786,20947171352106580284,223169444039365834362
%N A386956 a(n) = Sum_{k=0..n} (k+1) * 8^k * binomial(2*n+1,n-k).
%H A386956 Vincenzo Librandi, <a href="/A386956/b386956.txt">Table of n, a(n) for n = 0..1000</a>
%F A386956 a(n) = [x^n] 1/((1-9*x)^2 * (1-x)^n).
%F A386956 a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
%F A386956 a(n) = Sum_{k=0..n} (k+1) * 9^k * binomial(2*n-k-1,n-k).
%F A386956 G.f.: 2 * (1+sqrt(1-4*x))/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7)^2 ).
%t A386956 Table[Sum[(k+1) * 8^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 03 2025 *)
%o A386956 (PARI) a(n) = sum(k=0, n, (k+1)*8^k*binomial(2*n+1, n-k));
%o A386956 (Magma) [&+[(k+1) * 8^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 03 2025
%Y A386956 Cf. A088218, A258431, A384365, A386955.
%Y A386956 Cf. A386957.
%K A386956 nonn,changed
%O A386956 0,2
%A A386956 _Seiichi Manyama_, Aug 11 2025