A162507
Triangle read by rows, finite differences of an array generated by an infinite product (Cf. A162506).
Original entry on oeis.org
1, 2, 1, 2, 3, 1, 4, 3, 4, 1, 4, 9, 4, 5, 1, 6, 12, 12, 5, 6, 1, 6, 18, 24, 15, 6, 7, 1, 8, 24, 36, 30, 18, 7, 8, 1, 8, 33, 60, 60, 36, 21, 8, 9, 1, 10, 39, 88, 95, 72, 42, 24, 9, 10, 1, 10, 51, 124, 160, 138, 84, 48, 27, 10, 11
Offset: 2
The array =
1,...1,...1,...1,...1,...; = a
1,...1,...3,...3,...5,...; = a*b
1,...1,...3,...6,...8,...; = a*b*c
1,...1,...3,...6,..12,...; = a*b*c*d
...
taking finite differences from the top, then discarding the first "1",
we obtain triangle A162507:
1;
1, 2;
1, 2, 3;
1, 4, 3, 4;
1, 4, 9, 4, 5;
1, 6, 12, 12, 5, 6;
1, 6, 18, 24, 15, 6, 7;
1, 8, 24, 36, 30, 18, 7, 8;
1, 8, 33, 60, 60, 36, 21, 8, 9;
1, 10, 39, 88, 95, 72, 42, 24, 9, 10;
1, 10, 51, 124, 160, 138, 84, 48, 27, 10, 11;
...
A267007
Expansion of Product_{k>=1} (1 + (k-1)*x^k).
Original entry on oeis.org
1, 0, 1, 2, 3, 6, 8, 16, 20, 42, 51, 92, 132, 204, 299, 476, 644, 978, 1488, 2024, 3048, 4318, 6248, 8596, 12555, 17378, 24740, 34310, 47940, 65842, 93221, 125238, 173848, 239348, 324724, 445882, 602140, 816424, 1101096, 1495382, 1991892, 2684252, 3598248
Offset: 0
-
b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..42); # Alois P. Heinz, Aug 15 2019
-
nmax = 50; CoefficientList[Series[Product[1+(k-1)*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 0; Do[Do[poly[[j+1]] += (k-1)*poly[[j-k+1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly
A163318
Expansion of g.f.: Product_{k>=1} 1+k*x^k/(1-x^k)^2.
Original entry on oeis.org
1, 1, 4, 8, 19, 36, 76, 142, 272, 496, 900, 1592, 2784, 4792, 8138, 13688, 22703, 37380, 60838, 98310, 157298, 250162, 394332, 618032, 961512, 1487563, 2286610, 3496776, 5316666, 8044598, 12110538, 18147166, 27068692, 40203306, 59459998, 87587428, 128522850
Offset: 0
-
b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-1)*(j*i), j=1..n/i)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..40); # Alois P. Heinz, Feb 25 2013
-
terms = 40;
CoefficientList[Product[1 + k x^k/(1 - x^k)^2, {k, 1, terms}] + O[x]^terms, x] (* Jean-François Alcover, Nov 12 2020 *)
A346788
Product over all partitions lambda of n of the product of distinct parts in lambda.
Original entry on oeis.org
1, 1, 2, 6, 48, 1440, 414720, 2090188800, 1155790798848000, 226483217146419609600000, 302971317675145105975227187200000000, 37917003542135076706761224377027811868672000000000000, 45800346382799680410294841758069930049013501333211737122406400000000000000000
Offset: 0
-
a:= n-> mul(i, i=map(x-> {x[]}[], combinat[partition](n))):
seq(a(n), n=0..12);
-
a[n_] := Times @@ Times @@@ Union /@ IntegerPartitions[n];
a /@ Range[0, 20] (* Jean-François Alcover, Aug 09 2021 *)
-
a(n) = vecprod(apply(x->vecprod(Set(x)), partitions(n))); \\ Michel Marcus, Aug 04 2021
A369887
Sum of products of squares of parts , counted without multiplicity, in all partitions of n.
Original entry on oeis.org
1, 1, 5, 14, 34, 95, 208, 537, 1090, 2812, 5566, 12480, 26199, 53486, 112866, 229111, 450800, 885030, 1778190, 3319846, 6624376, 12354288, 23674929, 43485580, 81441398, 149864634, 273431081, 503205344, 906757150, 1630802024, 2920280596, 5166820832
Offset: 0
The partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1. So a(4) = 16 + 9 + 4 + 4 + 1 = 34.
A369888
Sum of products of cubes of parts , counted without multiplicity, in all partitions of n.
Original entry on oeis.org
1, 1, 9, 36, 108, 449, 1212, 4499, 10914, 43286, 103296, 306994, 867763, 2165484, 6627800, 16827227, 42203212, 104397436, 282967414, 632194758, 1809241372, 4120266946, 10256452121, 23140530512, 55030272918, 130803096050, 291295024121, 739011803928, 1634625423738
Offset: 0
The partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1. So a(4) = 64 + 27 + 8 + 8 + 1 = 108.
Showing 1-6 of 6 results.
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