Search: a356268 -id:a356268
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A356267
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a(n) = Sum_{k=0..n} binomial(2*n, k) * p(k), where p(k) is the partition function A000041.
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+10
4
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1, 3, 17, 97, 583, 3275, 18988, 104821, 584441, 3180889, 17295626, 92225785, 492811733, 2590911097, 13591889993, 70605682273, 365601169939, 1876312271003, 9605682510676, 48809295651049, 247315330613099, 1245888505795725, 6256686417801919, 31260996876796579
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) ~ erfc(Pi/(2*sqrt(6))) * 2^(2*n - 3) * exp(Pi*sqrt(2*n/3) + Pi^2/24) / (sqrt(3)*n).
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MATHEMATICA
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Table[Sum[Binomial[2*n, k] * PartitionsP[k], {k, 0, n}], {n, 0, 30}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A356281
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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).
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+10
4
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) ~ 2^(2*n - 1/2) * exp(3^(1/3) * Pi^(4/3) * n^(1/3) / 2^(8/3)) / sqrt(3*Pi*n).
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A356270
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a(n) = Sum_{k=0..n} binomial(2*k, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).
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+10
3
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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
(list;
graph;
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listen;
history;
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) ~ binomial(2*n,n) * q(n) * 4/3.
a(n) ~ 2^(2*n) * exp(Pi*sqrt(n/3)) / (3^(5/4) * sqrt(Pi) * n^(5/4)).
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MATHEMATICA
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Table[Sum[Binomial[2*k, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A356285
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a(n) = Sum_{k=0..n} binomial(3*n, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).
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+10
1
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1, 4, 22, 214, 1509, 12770, 107884, 874365, 6834843, 56722759, 463069914, 3666488610, 29512199193, 233492075573, 1858649112464, 14890457067926, 117154630898329, 917101099859767, 7257072314543086, 56653800922475280, 442687465112658972, 3467083846726752495
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) ~ 3^(3*n + 1/4) * exp(Pi*sqrt(n/3)) / (sqrt(Pi) * n^(5/4) * 2^(2*n + 2)).
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MATHEMATICA
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Table[Sum[Binomial[3*n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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