A110142 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.

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, 162, 18, 20, 9, 3840, 192, 144, 48, 30, 64, 16, 72, 21, 24, 50, 10, 1152, 240, 96, 56, 324, 36, 40, 18, 90, 96, 24, 28, 30, 11, 46080, 1536, 864, 288, 120, 256, 64, 144, 42, 48, 100

Offset: 1

Author

Paul D. Hanna, Jul 13 2005

Keywords

• nonn

Comments

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.

Examples

			Row 6 of A110141 is: {720,48,18,16,8,6,5,48,8,18,6};
divided respectively by: {6!,4!,3!,2!,2!,1!,1!,0!,0!,0!,0!}
with {4!,3!,2!,1!,0!} each occurring {1,1,2,2,4} times after 6!,
yields the initial A000041(6)=11 terms: {1,2,3,8,4,6,5,48,8,18,6}.
Sum of reciprocal terms at positions p(5)+1 through p(6) =
1/48 + 1/8 + 1/18 + 1/6 = 1-1+1/2!-1/3!+1/4!-1/5!+1/6!.
Other patterns emerge when the terms are read by groups
of terms in positions p(n-1)+1 through p(n):
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, 162,18,20,9;
3840,192,144,48,30,64,16, 72,21,24,50,10;
1152,240,96,56,324,36,40,18, 90,96,24,28,30,11;
46080,1536,864,288,120,256,64,144,42,48,100,20, 1944,108,60,27,384,32,35,72,12;
11520,1920,576,336,1296,144,160,72,180,192,48,56,60,22, 648,126,72,150,30,160,36,40,42,13; ...
		

Crossrefs

Cf. A110141, A110144, A000041, A002865.

Formula

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.