A356268
a(n) = Sum_{k=0..n} binomial(2*n, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).
Original entry on oeis.org
1, 3, 11, 62, 289, 1472, 7581, 38014, 184453, 918512, 4548393, 22077762, 107423503, 516720332, 2483445404, 11959145079, 57022343425, 270173627092, 1282971321633, 6047971597490, 28446033085527, 133714464665108, 625893086713686, 2919093380089383, 13596052503945537
Offset: 0
-
Table[Sum[Binomial[2*n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]
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.
-
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]
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
-
Table[Sum[Binomial[2*k, k] * PartitionsP[k], {k, 0, n}], {n, 0, 30}]
A356284
a(n) = Sum_{k=0..n} binomial(3*n, k) * p(k), where p(k) is the partition function A000041.
Original entry on oeis.org
1, 4, 37, 334, 3280, 29437, 282253, 2517904, 23209785, 206685325, 1858085653, 16266231810, 144339750406, 1250038867329, 10882952174845, 93546973843450, 804847296088574, 6843680884286307, 58300294406199829, 491683063753997014, 4148296662116385627, 34746182976196757434
Offset: 0
-
Table[Sum[Binomial[3*n, k] * PartitionsP[k], {k, 0, n}], {n, 0, 30}]
-
a(n) = sum(k=0, n, binomial(3*n, k)*numbpart(k)); \\ Michel Marcus, Aug 02 2022
Showing 1-4 of 4 results.