A356280
a(n) = Sum_{k=0..n} binomial(2*n, n-k) * p(k), where p(k) is the partition function A000041.
Original entry on oeis.org
1, 3, 12, 50, 211, 894, 3791, 16068, 68032, 287675, 1214761, 5122428, 21571028, 90718913, 381050570, 1598645263, 6699355413, 28044720813, 117281866330, 489999068614, 2045341248508, 8530263939665, 35547083083270, 148015639243691, 615870619714675, 2560734764460360
Offset: 0
- Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, VI.26. Catalan sums, p.417.
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Table[Sum[PartitionsP[k]*Binomial[2*n, n-k], {k, 0, n}], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[PartitionsP[k]*((1-2*x-Sqrt[1-4*x])/(2*x))^k / Sqrt[1-4*x], {k, 0, nmax}], {x, 0, nmax}], x]
A356283
a(n) = Sum_{k=0..n} binomial(3*n, n-k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).
Original entry on oeis.org
1, 4, 22, 131, 807, 5066, 32188, 206242, 1329733, 8614685, 56024538, 365491218, 2390613557, 15671221522, 102925324569, 677110860689, 4460956827127, 29427611146335, 194348311824025, 1284856925961827, 8502252246841668, 56309476194587377, 373220349572126265
Offset: 0
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Table[Sum[PartitionsQ[k]*Binomial[3*n, n-k], {k, 0, n}], {n, 0, 30}]
A356289
a(n) = Sum_{k=0..n} binomial(2*n, n-k) * v(k), where v(k) is the number of overpartitions of n (A015128).
Original entry on oeis.org
1, 4, 18, 82, 372, 1676, 7500, 33358, 147570, 649722, 2848524, 12441434, 54155774, 235008672, 1016971480, 4389589484, 18902538548, 81222609020, 348308661820, 1490884718484, 6370468593732, 27176620756392, 115760526170340, 492386739902574, 2091554077819948, 8873225318953248
Offset: 0
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Table[Sum[Sum[PartitionsP[k-j]*PartitionsQ[j], {j, 0, k}] * Binomial[2*n, n-k], {k, 0, n}], {n, 0, 30}]
A356290
a(n) = Sum_{k=0..n} binomial(3*n, n-k) * v(k), where v(k) is the number of overpartitions of n (A015128).
Original entry on oeis.org
1, 5, 31, 200, 1309, 8627, 57082, 378648, 2516111, 16740913, 111494801, 743137984, 4956359312, 33074272702, 220810039566, 1474764797488, 9853307017341, 65853733243281, 440255398634199, 2944041287677060, 19691951641479427, 131744163990056479, 881586559906575688
Offset: 0
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Table[Sum[Sum[PartitionsP[k-j]*PartitionsQ[j], {j, 0, k}] * Binomial[3*n, n-k], {k, 0, n}], {n, 0, 30}]
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