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%I #5 Mar 30 2012 18:36:49
%S 1,2,3,8,4,6,5,48,8,18,6,24,10,12,7,384,32,36,12,15,32,8,144,40,24,14,
%T 162,18,20,9,3840,192,144,48,30,64,16,72,21,24,50,10,1152,240,96,56,
%U 324,36,40,18,90,96,24,28,30,11,46080,1536,864,288,120,256,64,144,42,48,100
%N Limit of rows of triangle A110141 after dividing respectively by a list of factorials, with (n-j-1)! repeated A002865(j+1) times in the list as j=1..n-1.
%C Row n of triangle A110141 lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k). A002865 equals the first differences of the partition numbers. A110144 lists terms at positions p(n)+1.
%F a(p(n)) = n where p(n) = A000041(n) (partition numbers) for n>=1. Sum_{k=p(n-1)+1..p(n)} 1/a(k) = Sum_{k=0..n} (-1)^k/k!, for n>1.
%e Row 6 of A110141 is: {720,48,18,16,8,6,5,48,8,18,6};
%e divided respectively by: {6!,4!,3!,2!,2!,1!,1!,0!,0!,0!,0!}
%e with {4!,3!,2!,1!,0!} each occurring {1,1,2,2,4} times after 6!,
%e yields the initial A000041(6)=11 terms: {1,2,3,8,4,6,5,48,8,18,6}.
%e Sum of reciprocal terms at positions p(5)+1 through p(6) =
%e 1/48 + 1/8 + 1/18 + 1/6 = 1-1+1/2!-1/3!+1/4!-1/5!+1/6!.
%e Other patterns emerge when the terms are read by groups
%e of terms in positions p(n-1)+1 through p(n):
%e 1;
%e 2;
%e 3;
%e 8,4;
%e 6, 5;
%e 48,8, 18,6;
%e 24,10, 12,7;
%e 384,32,36,12, 15,32,8;
%e 144,40,24,14, 162,18,20,9;
%e 3840,192,144,48,30,64,16, 72,21,24,50,10;
%e 1152,240,96,56,324,36,40,18, 90,96,24,28,30,11;
%e 46080,1536,864,288,120,256,64,144,42,48,100,20, 1944,108,60,27,384,32,35,72,12;
%e 11520,1920,576,336,1296,144,160,72,180,192,48,56,60,22, 648,126,72,150,30,160,36,40,42,13; ...
%Y Cf. A110141, A110144, A000041, A002865.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 13 2005
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