cp's OEIS Frontend

A386986 a(n) = Sum_{k=0..n} (k+1) * 8^k * binomial(2*n+2,n-k).

%I A386986 #25 Aug 19 2025 04:17:48
%S A386986 1,20,303,4088,51730,628488,7423899,85904688,978506478,11008191800,
%T A386986 122603713078,1354213651728,14854030654372,161966063719712,
%U A386986 1757042561230515,18976059641899872,204140891541240918,2188510439907779064,23389705325379996834,249285017279237071440
%N A386986 a(n) = Sum_{k=0..n} (k+1) * 8^k * binomial(2*n+2,n-k).
%H A386986 Vincenzo Librandi, <a href="/A386986/b386986.txt">Table of n, a(n) for n = 0..400</a>
%F A386986 a(n) = [x^n] 1/((1-9*x)^2 * (1-x)^(n+1)).
%F A386986 a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+2,k) * binomial(2*n-k,n-k).
%F A386986 a(n) = Sum_{k=0..n} (k+1) * 9^k * binomial(2*n-k,n-k).
%F A386986 G.f.: 4/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7)^2 ).
%F A386986 a(n) ~ 7 * n * 3^(4*n+2) / 2^(3*n+6). - _Vaclav Kotesovec_, Aug 12 2025
%F A386986 D-finite with recurrence 520*n*a(n) +(-8641*n-1633)*a(n-1) +486*(81*n-32)*a(n-2) +26244*(-2*n+3)*a(n-3)=0. - _R. J. Mathar_, Aug 19 2025
%t A386986 Table[Sum[(k+1)* 8^k*Binomial[2*n+2,n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 14 2025 *)
%o A386986 (PARI) a(n) = sum(k=0, n, (k+1)*8^k*binomial(2*n+2, n-k));
%o A386986 (Magma) [&+[(k+1)*8^k * Binomial(2*n+2, n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 14 2025
%Y A386986 Cf. A386957, A386958.
%Y A386986 Cf. A000984, A002457, A383832.
%K A386986 nonn
%O A386986 0,2
%A A386986 _Seiichi Manyama_, Aug 12 2025