A379025
a(n) = Sum_{k=0..n} binomial(3*n+k-1,k) * binomial(3*n+k,n-k).
Original entry on oeis.org
1, 6, 78, 1149, 17850, 285711, 4661727, 77086008, 1287322866, 21661521945, 366687839133, 6237631866417, 106535632157643, 1825763898882189, 31379978657609100, 540688387589377764, 9336602657251874754, 161534120354250452361, 2799488717098336992687
Offset: 0
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[&+[Binomial(3*n+k-1, k)*Binomial(3*n+k, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Dec 21 2024
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Table[Sum[Binomial[3*n+k-1,k]*Binomial[3*n+k, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Dec 21 2024 *)
a[n_]:= Binomial[3*n, n]*HypergeometricPFQ[{-n, 3*n, 1 + 3*n}, {1/2 + n, 1 + n}, -1/4]; Array[a,19,0] (* Stefano Spezia, Dec 22 2024 *)
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a(n) = sum(k=0, n, binomial(3*n+k-1, k)*binomial(3*n+k, n-k));
A379021
Expansion of (1/x) * Series_Reversion( x * ((1 - x - x^2)/(1 + x))^2 ).
Original entry on oeis.org
1, 4, 26, 206, 1813, 17032, 167287, 1697044, 17643322, 186997570, 2012973499, 21948003052, 241883091289, 2690117648372, 30153678822007, 340305271736134, 3863616751855069, 44097785533620550, 505692279260755753, 5823592506326814874, 67320958983831426221
Offset: 0
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a(n) = 2*sum(k=0, n, binomial(2*n+k+2, k)*binomial(2*n+k+2, n-k)/(2*n+k+2));
A379026
a(n) = Sum_{k=0..n} binomial(4*n+k-1,k) * binomial(4*n+k,n-k).
Original entry on oeis.org
1, 8, 136, 2612, 52888, 1103248, 23458756, 505519792, 11001461560, 241240165796, 5321735043496, 117969960106380, 2625673485660100, 58638653062716488, 1313363972969179904, 29489827322243843032, 663600214363813934328, 14961465721142755457484
Offset: 0
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a(n) = sum(k=0, n, binomial(4*n+k-1, k)*binomial(4*n+k, n-k));
A379085
a(n) = Sum_{k=0..floor(n/3)} binomial(2*n+k-1,k) * binomial(2*n+k,n-3*k).
Original entry on oeis.org
1, 2, 6, 26, 142, 802, 4434, 24222, 132686, 733076, 4081926, 22853052, 128427106, 723862856, 4090573570, 23170106086, 131515806574, 747875338152, 4259810283828, 24298797944956, 138787172202182, 793651842511512, 4543393775520936, 26035130683198684, 149325002408646002
Offset: 0
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a(n) = sum(k=0, n\3, binomial(2*n+k-1, k)*binomial(2*n+k, n-3*k));
A379084
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k-1,k) * binomial(2*n+k,n-2*k).
Original entry on oeis.org
1, 2, 10, 62, 394, 2562, 16966, 113794, 770458, 5254658, 36046470, 248449104, 1719175846, 11935608518, 83100064834, 579994824042, 4056746450106, 28428354905268, 199550820571858, 1402832286126650, 9875127071717694, 69599814539512900, 491081313666879968, 3468458841769675496
Offset: 0
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a(n) = sum(k=0, n\2, binomial(2*n+k-1, k)*binomial(2*n+k, n-2*k));
Showing 1-5 of 5 results.