A382537
Expansion of 1/(1 - x*(1 + 4*x)^(5/2)).
Original entry on oeis.org
1, 1, 11, 51, 211, 1061, 4923, 22765, 107687, 502479, 2352231, 11022911, 51590795, 241559783, 1131156175, 5295875131, 24797055115, 116104311885, 543622665219, 2545347081565, 11917847333151, 55801588711565, 261274518155435, 1223337818786305, 5727913381451455
Offset: 0
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R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + 4*x)^(5/2)); seq := [ Coefficient(f, n) : n in [0..30] ];seq; // Vincenzo Librandi, Apr 02 2025
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Table[Sum[4^(n-k)*Binomial[5*k/2,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Apr 02 2025 *)
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a(n) = sum(k=0, n, 4^(n-k)*binomial(5*k/2, n-k));
A382538
Expansion of 1/(1 - x*(1 + 4*x)^(7/2)).
Original entry on oeis.org
1, 1, 15, 99, 519, 3165, 19503, 115053, 688803, 4141863, 24778355, 148376447, 889216143, 5326274463, 31903872267, 191123789739, 1144894457103, 6858232252437, 41083285178247, 246102886383661, 1474237118571467, 8831178384769525, 52901735792001759
Offset: 0
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R:=PowerSeriesRing(Rationals(), 28); Coefficients(R!( 1/(1 - x*(1 + 4*x)^(7/2)))); // Vincenzo Librandi, May 16 2025
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Table[Sum[4^(n-k)* Binomial[7*k/2, n-k],{k,0,n}],{n,0,28}] (* Vincenzo Librandi, May 16 2025 *)
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a(n) = sum(k=0, n, 4^(n-k)*binomial(7*k/2, n-k));
A382649
Expansion of 1/(1 - x*(1 + 4*x)^(3/2))^2.
Original entry on oeis.org
1, 2, 15, 52, 213, 834, 3043, 11576, 41601, 152458, 544039, 1950132, 6895773, 24403302, 85542339, 300101048, 1044436937, 3639851814, 12594713911, 43660404108, 150357976533, 518991977194, 1780132570723, 6122965091976, 20928650616113, 71779065646510, 244590689773839
Offset: 0
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R:=PowerSeriesRing(Rationals(), 28); Coefficients(R!( 1/(1 - x*(1 + 4*x)^(3/2))^2)); // Vincenzo Librandi, May 13 2025
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Table[Sum[4^(n-k)* (k+1)* Binomial[3*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(3*k/2, n-k));
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));
Showing 1-4 of 4 results.
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