A356270
a(n) = Sum_{k=0..n} binomial(2*k, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).
Original entry on oeis.org
1, 3, 9, 49, 189, 945, 4641, 21801, 99021, 487981, 2335541, 10800725, 51363065, 238573865, 1121139065, 5309312105, 24543884585, 113220920945, 530677144745, 2439321389945, 11261499234425, 52169097691865, 239433905462945, 1095710701133345, 5029918350471545
Offset: 0
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Table[Sum[Binomial[2*k, 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
Original entry on oeis.org
1, 2, 6, 21, 91, 385, 1837, 8272, 39732, 185592, 891024, 4183040, 20199964, 95232864, 456282264, 2162574984, 10330196754, 48834699384, 232725598884, 1098684561984, 5214388065324, 24591671545164, 116257200312444, 546797015443194, 2578396047478494, 12098087101521510
Offset: 0
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Table[Sum[CatalanNumber[k]*PartitionsP[k], {k, 0, n}], {n, 0, 30}]
Showing 1-3 of 3 results.