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A063834 Twice partitioned numbers: the number of ways a number can be partitioned into not necessarily different parts and each part is again so partitioned. 231
1, 1, 3, 6, 15, 28, 66, 122, 266, 503, 1027, 1913, 3874, 7099, 13799, 25501, 48508, 88295, 165942, 299649, 554545, 997281, 1817984, 3245430, 5875438, 10410768, 18635587, 32885735, 58399350, 102381103, 180634057, 314957425, 551857780, 958031826, 1667918758 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
These are different from plane partitions.
For ordered partitions of partitions see A055887 which may be computed from A036036 and A048996. - Alford Arnold, May 19 2006
Twice partitioned numbers correspond to triangles (or compositions) in the multiorder of integer partitions. - Gus Wiseman, Oct 28 2015
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..5000 (terms n=1..500 from T. D. Noe)
FORMULA
G.f.: 1/Product_{k>0} (1-A000041(k)*x^k). n*a(n) = Sum_{k=1..n} b(k)*a(n-k), a(0) = 1, where b(k) = Sum_{d|k} d*A000041(d)^(k/d) = 1, 5, 10, 29, 36, 110, 106, ... . - Vladeta Jovovic, Jun 19 2003
From Vaclav Kotesovec, Mar 27 2016: (Start)
a(n) ~ c * 5^(n/4), where
c = 96146522937.7161898848278970039269600938032826... if n mod 4 = 0
c = 96146521894.9433858914667933636782092683849082... if n mod 4 = 1
c = 96146522937.2138934755566928890704687838407524... if n mod 4 = 2
c = 96146521894.8218716328341714149619262713426755... if n mod 4 = 3
(End)
EXAMPLE
G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + 266*x^8 + ...
If n=6, a possible first partitioning is (3+3), resulting in the following second partitionings: ((3),(3)), ((3),(2+1)), ((3),(1+1+1)), ((2+1),(3)), ((2+1),(2+1)), ((2+1),(1+1+1)), ((1+1+1),(3)), ((1+1+1),(2+1)), ((1+1+1),(1+1+1)).
MAPLE
with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
b(n, i-1)+`if`(i>n, 0, numbpart(i)*b(n-i, i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Alois P. Heinz, Nov 26 2015
MATHEMATICA
Table[Plus @@ Apply[Times, IntegerPartitions[i] /. i_Integer :> PartitionsP[i], 2], {i, 36}]
(* second program: *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i > n, 0, PartitionsP[i]*b[n-i, i]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 19 2016 */
CROSSREFS
Row sums of A321449.
Column k=2 of A323718.
Cf. A327769.
Sequence in context: A318396 A034953 A086737 * A139117 A226736 A066708
KEYWORD
nonn,nice
AUTHOR
Wouter Meeussen, Aug 21 2001
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Nov 26 2015
STATUS
approved

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Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)