A356267
a(n) = Sum_{k=0..n} binomial(2*n, k) * p(k), where p(k) is the partition function A000041.
Original entry on oeis.org
1, 3, 17, 97, 583, 3275, 18988, 104821, 584441, 3180889, 17295626, 92225785, 492811733, 2590911097, 13591889993, 70605682273, 365601169939, 1876312271003, 9605682510676, 48809295651049, 247315330613099, 1245888505795725, 6256686417801919, 31260996876796579
Offset: 0
-
Table[Sum[Binomial[2*n, k] * PartitionsP[k], {k, 0, n}], {n, 0, 30}]
A356281
a(n) = Sum_{k=0..n} binomial(2*n, n-k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).
Original entry on oeis.org
1, 3, 11, 43, 172, 695, 2823, 11501, 46940, 191791, 784148, 3207196, 13119733, 53670793, 219545353, 897957702, 3672093558, 15013596535, 61370565546, 250803861369, 1024716136043, 4185683293934, 17093143284723, 69786349712519, 284847779542644, 1162385753008079
Offset: 0
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Table[Sum[PartitionsQ[k]*Binomial[2*n, n-k], {k, 0, n}], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k]*((1-2*x-Sqrt[1-4*x])/(2*x))^k / Sqrt[1-4*x], {k, 0, nmax}], {x, 0, nmax}], x]
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
-
Table[Sum[Binomial[2*k, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]
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
-
Table[Sum[Binomial[3*n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]
Showing 1-4 of 4 results.