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Number of compositions (ordered partitions) of n into distinct parts such that the smallest part is equal to the number of parts.
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%I #8 Jan 15 2022 21:49:18

%S 1,0,0,0,2,2,2,2,2,2,2,8,8,14,14,20,20,26,26,32,32,62,62,92,116,146,

%T 170,224,248,302,350,404,452,530,698,776,968,1166,1478,1700,2132,2474,

%U 3050,3512,4208,4814,5750,6476,7556,8522,10562,11672,13952,16022,19286,22316,26540

%N Number of compositions (ordered partitions) of n into distinct parts such that the smallest part is equal to the number of parts.

%H Seiichi Manyama, <a href="/A339446/b339446.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F G.f.: Sum_{k>=1} k! * x^(k*(3*k - 1)/2) / Product_{j=1..k-1} (1 - x^j).

%e a(12) = 8 because we have [10, 2], [2, 10], [5, 4, 3], [5, 3, 4], [4, 5, 3], [4, 3, 5], [3, 5, 4] and [3, 4, 5].

%t nmax = 60; CoefficientList[Series[Sum[k! x^(k (3 k - 1)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%Y Cf. A006141, A032020, A096401, A098133.

%K nonn

%O 1,5

%A _Ilya Gutkovskiy_, Dec 05 2020