A292463
Number of partitions of n with n kinds of 1.
Original entry on oeis.org
1, 1, 4, 14, 51, 188, 702, 2644, 10026, 38223, 146359, 562456, 2168134, 8379539, 32459199, 125984039, 489837300, 1907490728, 7438346255, 29042470132, 113522618066, 444199913556, 1739735079466, 6819657196928, 26753893533257, 105034060120469, 412637434996367
Offset: 0
a(2) = 4: 2, 1a1a, 1a1b, 1b1b.
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b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))
end:
a:= n-> b(n$3):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
(numtheory[sigma](j)+k-1)*b(n-j, k), j=1..n)/n)
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=1,
combinat[numbpart](n), b(n-1, k) +b(n, k-1)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
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Table[SeriesCoefficient[1/(1-x)^(n-1) * Product[1/(1-x^k), {k,1,n}], {x,0,n}], {n,0,30}] (* Vaclav Kotesovec, Sep 19 2017 *)
A292613
a(n) = [x^n] 1/(1-x)^n * Product_{k=1..n} 1/(1-x^k).
Original entry on oeis.org
1, 2, 7, 25, 92, 343, 1292, 4902, 18703, 71677, 275694, 1063636, 4114131, 15948762, 61946290, 241013869, 939125870, 3664299332, 14314777054, 55982787136, 219158088711, 858728875776, 3367576480747, 13216392846128, 51905939548950, 203989227456894, 802164259099114
Offset: 0
Illustration of comment for n=3, a(3)=25 :
Among the 11 integer partitions of 6, 3 have at least 3 ones.
3,1,1,1 ; 2,1,1,1,1; 1,1,1,1,1,1;
There are respectively 1, 4 and 20 ways to pick 3 of these.
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Table[SeriesCoefficient[1/(1-x)^n*Product[1/(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]
A292617
Convolution of number of partitions into distinct parts and Catalan numbers.
Original entry on oeis.org
1, 2, 4, 10, 25, 70, 209, 656, 2137, 7155, 24447, 84864, 298374, 1060151, 3800365, 13727145, 49910870, 182522747, 670896855, 2477250003, 9184502747, 34177467134, 127606759053, 477890336663, 1794697782990, 6757164079051, 25501212956975, 96450275088260
Offset: 0
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Table[Sum[PartitionsQ[k]*CatalanNumber[n-k], {k, 0, n}], {n, 0, 50}]
A304824
Convolution of central binomial coefficients and partition numbers.
Original entry on oeis.org
1, 3, 10, 33, 113, 397, 1431, 5249, 19514, 73260, 277100, 1054248, 4029859, 15463765, 59531725, 229816430, 889301153, 3448417251, 13396337155, 52126461984, 203124067675, 792559645912, 3096104725974, 12107810534937, 47395948167885, 185697860476576
Offset: 0
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Table[Sum[PartitionsP[n-k]*Binomial[2*k, k], {k, 0, n}], {n, 0, 25}]
A292619
Convolution of number of overpartitions and Catalan numbers.
Original entry on oeis.org
1, 3, 8, 21, 54, 144, 404, 1195, 3712, 12000, 39994, 136400, 473430, 1665868, 5926476, 21275805, 76964808, 280250088, 1026309908, 3777411342, 13965286180, 51837285776, 193107846304, 721732334136, 2705480787422, 10169387310362, 38320472420462, 144733083435688
Offset: 0
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Table[Sum[Sum[PartitionsP[k-j] * PartitionsQ[j], {j, 0, k}] * CatalanNumber[n-k], {k, 0, n}], {n, 0, 50}]
Showing 1-5 of 5 results.
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