cp's OEIS Frontend

A365554 Number of increasing paths from the bottom to the top of the n-hypercube (as a graded poset) which first encounter a vector of isolated zeros at stage k, weighted by k.

%I A365554 #60 Feb 24 2024 15:31:44
%S A365554 2,10,60,396,2976,25056,234720,2423520,27371520,335819520,4449150720,
%T A365554 63318931200,963548006400,15614378035200,268480048435200,
%U A365554 4882321001779200,93627018326016000,1888394741194752000,39963486306078720000,885457095215616000000
%N A365554 Number of increasing paths from the bottom to the top of the n-hypercube (as a graded poset) which first encounter a vector of isolated zeros at stage k, weighted by k.
%C A365554 These are the numerators in calculating an expected value. The expectation of the number of steps one takes in marking the elements of a predetermined list before reaching a state where only isolated unmarked entries remain.
%F A365554 a(n) = Sum_{k=floor(n/2)..n-1} k*(binomial(k+1,n-k)-binomial(k-1,n-k))*k!*(n-k)!.
%e A365554 For n=5, an example vector of isolated 0's is 01011, which has k=3 1's.
%e A365554 For n=3, the following paths (from 000 to 111) reach isolated 0's at k=1 many 1's (010):
%e A365554   000,010,011,111
%e A365554   000,010,110,111
%e A365554 The following paths reach isolated 0's only at k=2 1's:
%e A365554   000,100,110,111
%e A365554   000,100,101,111
%e A365554   000,001,101,111
%e A365554   000,001,011,111
%e A365554 So 2 paths of k=1 and 4 paths of k=2 are weighted total a(3) = 2*1 + 4*2 = 10.
%o A365554 (SageMath)
%o A365554 k, n = var('k,n')
%o A365554 sum((binomial(k+1,n-k)-binomial(k-1,n-k))*factorial(k)*factorial(n-k), k, floor(n/2),n-1)
%o A365554 (PARI) a(n) = sum(k=n\2, n-1, k*(binomial(k+1,n-k)-binomial(k-1,n-k))*k!*(n-k)!) \\ _Andrew Howroyd_, Feb 23 2024
%Y A365554 Cf. A067331.
%K A365554 nonn
%O A365554 2,1
%A A365554 _Brian Darrow, Jr._ and Joe Fields, Feb 20 2024