A382650
Expansion of 1/(1 - x*(1 + 4*x)^(3/2))^3.
Original entry on oeis.org
1, 3, 24, 100, 471, 2043, 8422, 34818, 137649, 543655, 2096508, 8031948, 30355155, 113929497, 423562614, 1565841650, 5745557853, 20989365057, 76206968356, 275721399480, 992423144247, 3562075121911, 12728422443654, 45379998032202, 161158522838105, 571293893581389
Offset: 0
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a(n) = sum(k=0, n, 4^(n-k)*binomial(k+2, 2)*binomial(3*k/2, n-k));
A382647
Expansion of 1/(1 - x*(1 + 4*x)^(1/2))^2.
Original entry on oeis.org
1, 2, 7, 12, 37, 50, 187, 128, 1057, -502, 7679, -14420, 73453, -212554, 843019, -2848064, 10602409, -37875706, 139533151, -510006524, 1885309253, -6974175142, 25940881947, -96731191728, 361980829841, -1358121976978, 5109416286295, -19267391982612
Offset: 0
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R:=PowerSeriesRing(Rationals(), 28); Coefficients(R!( 1/(1 - x*(1 + 4*x)^(1/2))^2)); // Vincenzo Librandi, May 13 2025
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Table[Sum[4^(n-k)* (k+1)* Binomial[k/2, n-k],{k,0,n}],{n,0,28}] (* Vincenzo Librandi, May 13 2025 *)
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a(n) = sum(k=0, n, 4^(n-k)*(k+1)*binomial(k/2, n-k));
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