OFFSET
0,1
COMMENTS
The column sequences can also be calculated using sequences which map to associated partitions. For example, 4 32 132 392 ... maps to 5+5+5+4 (n=19) and sequence 5 50 245 840 ... maps to 4+4+4+4+3. Many partitions map to the same sequences since the mapping depends only on the "degree" of the partition. In the above two cases, the degrees are 31 and 41 respectively. At n = 20 the relevant degrees are: 21,31,211,311,22,221,42,212,321,24 and 61. The associated partitions can be permuted with the number of ways as indicated: 3 4 12 20 6 30 15 30 60 15 and 7 ways. Adding these values with the 32 and 50 ways from our first two sequences confirms that A110553(2) = 284.
LINKS
Georg Fischer, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14, -91, 364, -1001, 2002, -3003, 3432, -3003, 2002, -1001, 364, -91, 14, -1).
FORMULA
G.f.: 9+284*x+3004*x^2 -x^3*(-19078 +178154*x -826578*x^2 +2465215*x^3 -5191980*x^4 +8073520*x^5 -9475220*x^6 +8461596*x^7 -5732830*x^8 +2904174*x^9 -1067563*x^10 +269335*x^11 -41760*x^12 +3003*x^13) /(x-1)^14. - R. J. Mathar, Aug 28 2018
EXAMPLE
An examination of the relevant ordered Gaussian polynomials reveals the following distributions:
5 4
7 120 120 34 3
112 1127 1190 470 96 9
882 6692 7147 3270 910 162 15
therefore the sequence begins
9
284
3004
19078
...
MATHEMATICA
[LinearRecurrence[{14, -91, 364, -1001, 2002, -3003, 3432, -3003, 2002, -1001, 364, -91, 14, -1}, {9, 284, 3004, 19078, 88938, 335612, 1084387, 3109060, 8104089, 19539904, 44141520, 94346102, 192252586, 375787005, 708083995, 1291443529, 2287680232}, 1001]] (* Georg Fischer, Feb 28 2019_ *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alford Arnold, Jul 29 2005
EXTENSIONS
More terms from R. J. Mathar, Aug 28 2018
STATUS
approved