A356285
a(n) = Sum_{k=0..n} binomial(3*n, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).
Original entry on oeis.org
1, 4, 22, 214, 1509, 12770, 107884, 874365, 6834843, 56722759, 463069914, 3666488610, 29512199193, 233492075573, 1858649112464, 14890457067926, 117154630898329, 917101099859767, 7257072314543086, 56653800922475280, 442687465112658972, 3467083846726752495
Offset: 0
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Table[Sum[Binomial[3*n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]
A356286
a(n) = Sum_{k=0..n} binomial(3*k, k) * p(k), where p(k) is the partition function A000041.
Original entry on oeis.org
1, 4, 34, 286, 2761, 23782, 227986, 1972186, 18152548, 158757298, 1420647928, 12258704248, 108637887148, 929002856992, 8065133782792, 68761800685576, 589631899738033, 4976639418495358, 42293283621258283, 354415428588891283, 2982701933728936648, 24857294772400460368
Offset: 0
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Table[Sum[Binomial[3*k, k] * PartitionsP[k], {k, 0, n}], {n, 0, 30}]
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a(n) = sum(k=0, n, binomial(3*k, k)*numbpart(k)); \\ Michel Marcus, Aug 02 2022
A356287
a(n) = Sum_{k=0..n} binomial(3*k, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).
Original entry on oeis.org
1, 4, 19, 187, 1177, 10186, 84442, 665842, 5078668, 42573268, 343023418, 2665464058, 21440629558, 167644287550, 1330569327310, 10641989818078, 82797155054782, 644097780350332, 5102709814966162, 39499844158337962, 307777892529889642, 2406854983109480302
Offset: 0
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Table[Sum[Binomial[3*k, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]