A356269
a(n) = Sum_{k=0..n} binomial(2*k, k) * p(k), where p(k) is the partition function A000041.
Original entry on oeis.org
1, 3, 15, 75, 425, 2189, 12353, 63833, 346973, 1805573, 9565325, 49069517, 257289529, 1307750129, 6723491129, 34024174649, 172873744739, 865954792079, 4359881882579, 21679061144579, 108108834714719, 534409071271199, 2642716232918639, 12975671796056639, 63765647596939139
Offset: 0
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Table[Sum[Binomial[2*k, k] * PartitionsP[k], {k, 0, n}], {n, 0, 30}]
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}]
Original entry on oeis.org
1, 2, 4, 14, 42, 168, 696, 2841, 11421, 50317, 218277, 923709, 4043889, 17416089, 76253769, 338014584, 1469460024, 6395962044, 28367342244, 123799554504, 543903261384, 2403339554904, 10545287718864, 46223487538464, 203591793511992, 893988182518176, 3924601439423256
Offset: 0
-
Table[Sum[CatalanNumber[k]*PartitionsQ[k], {k, 0, n}], {n, 0, 30}]
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