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%I #8 Aug 28 2018 09:56:40
%S 1,64,731,4553,20155,71272,214653,572743,1389702,3122752,6585183,
%T 13162741,25131718,46115029,81722067,140429357,234772177,382932581,
%U 610826859,954815625,1465182669,2210554686,3283463257,4807283267,6944818576,9908846494,13974977743,19497238421,26926835328
%N Column 10 of array illustrated in A089574 and related to A034261.
%C A109820 can be decomposed into 30 sequences. These 30 associated sequences can be inferred from the 30 ways of partitioning the number nine: 9 81 72 63 54 ... the complete listing is available in the Handbook of Mathematical Functions (1964) p. 831. Consider, for example, the three ways of partitioning the number three: 3, 21 and 111; prepend each partition then add one to each value - yielding 44, 332 and 2222. These "associated" partitions are then used to derive the associated sequences. 44 => A000330, 332 => A006011 and 2222 => A034263. Summing these three sequences yields A089574.
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
%F G.f. 1+64*x -x^2*(-731 +4219*x -13765*x^2 +30910*x^3 -49804*x^4 +58458*x^5 -50237*x^6 +31394*x^7 -13931*x^8 +4171*x^9 -757*x^10 +63*x^11)/(x-1)^12 . - _R. J. Mathar_, Aug 28 2018
%e a(1) = 1 because the only associated partition 4444 for n = 16 cannot be permuted.
%e a(2) = 64 because the associated partitions can be permuted in 3 + 4 + 12 + 9 + 20 + 10 + 6 ways when n = 17.
%Y Cf. A000330 (column 2), A086602 (column 3), A089574 (column 4), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13)
%K easy,nonn
%O 0,2
%A _Alford Arnold_, Jul 05 2005
%E Extended by _R. J. Mathar_, Aug 28 2018
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