cp's OEIS Frontend

This sequence arises when evaluating the generalized sub-volumes of the linearly weighted (n-1)-simplex in dimension n-1. For instance, in dimension 1 where n=2, the 1-simplex is the interval [H;J] of the real line (we suppose H < J). When H is weighted by the real h and J by j, the signed surface of the polygon {(H,0),(J,0),(J,j),(H,h)} of the Euclidean plane is S = (h+j)/2*(J-H).

Then we consider I the middle of [H;J]. It is linearly weighted by i = (h+j)/2. When we search for the weights w1(2) and w2(2) so that the 2 equations 2*Sh/(J-H) = h*w1(2) + j*w2(2) = (h+i)/2 and 2*Sj/(J-H) = h*w2(2) + j*w1(2) = (i+j)/2 are verified (which implies Sh + Sj = S also), we find that w1(2) = a(2)/A098916(2) = 3/4 and w2(2) = A067318(2)/A098916(2) = 1/4.

Even in higher dimensions (n > 2), there are only 2 weights: one for the considered sub-volume and the other for the other sub-volumes. For instance, in dimension 2 where n=3, the first weight w1(3) = 11/18 refers to the part of the triangle which is delimited by the 4 points: one top A, then the middle of [A;B], then the center of gravity, then the middle of [A;C]; and w2(3) = 7/36 refers to any of the 2 other parts of the triangle.

The first nonzero column gives the factorial numbers, which are Stirling_1(*,1), the rightmost diagonal gives Stirling_1(*,2), so this triangle may be regarded as interpolating between the first two columns of the Stirling numbers of the first kind.

This is a slice (the right-hand wall) through the infinite square pyramid described in the link. The other three walls give A007318 and A008276 (twice).

The coefficients of the A165674 triangle are generated by the asymptotic expansion of the higher order exponential integral E(x,m=2,n). The a(n) formulas for the coefficients in the right hand columns of this triangle lead to Wiggen's triangle A028421 and their o.g.f.s. lead to the sequence given above. Some right hand columns of the A165674 triangle are A080663, A165676, A165677, A165678 and A165679. - Johannes W. Meijer, Oct 07 2009

Row sums of A134433 starting from k=3.

a(n) = sum( (-1)^k*k*A008276(n,k), k=1..n-1).

a(n) = sum( (-1)^k*k*A054654(n,k), k=1..n-2).

For n >= 3, a(n) = number whose factorial base representation (A007623) begins with digits {n-1} and {n-2} followed by n-3 zeros. Viewed in that base, this sequence looks like this: 1, 21, 320, 4300, 54000, 650000, 7600000, 87000000, 980000000, A900000000, BA000000000, ... (where "digits" A and B stand for placeholder values 10 and 11 respectively). - Antti Karttunen, May 07 2015.

Sequences A067369, A067370 and A067318 are related: A067318 = A067369 + A067370. A067318 counts transpositions in the symmetric group, denoted S_n. One can think of the transpositions in S_n as being split between the alternating group A_n and its complement, which we call the periphery and denote P_N. For n >= 3, A067369 v(P_N) and A067370 v(A_n) always differ by (n-2)!. When n is odd, v(A_n) is larger; when n is even, v(P_N) is larger. This gives new meaning to the name alternating group. The average weight of permutation in A_n converges with the average weight for a permutation in P_N at infinity.

Sequences A067369, A067370 and A067318 are related. A067318 counts transpositions in the symmetric group, denoted S_n. One can think of the transpositions in S_n as being split between the alternating group A_n and its complement, which we call the periphery and denote P_N. For n >= 3, A067369 v(P_N) and A067370 v(A_n) always differ by (n-2)!. When n is odd, v(A_n) is larger; when n is even, v(P_N) is larger. This gives new meaning to the name alternating group. The average weight of a permutation in A_n converges with the average weight for a permutation in P_N at infinity.

Triangle of coefficients of polynomials P[n]. Let F(t) satisfy dF/dt = exp(x*(exp(F)-1)) and F(0)=0. Then F(t) = Sum_{n>=0} P[n]/n! t^n, where P[n] is a polynomial in x of degree n-1. The constant term of the polynomial is zero for n >= 2. The coefficient of x is 1 for n >= 2. The coefficient of x^n in P[n+1] is n!. The value at 1 is given by sequence A007549.

w(p) is defined (by Edelman, Simion and White) in the following way: if p = (c[1])(c[2])... is expressed in standard cycle form (i.e., cycles ordered by increasing smallest elements with each cycle written with its smallest element in the first position), then w(p) = 0*|c[1]| + 1*|c[2]| + 2*|c[3]| + ..., where |c[j]| denotes the number of entries in the cycle c[j].

Row n has 1 + n*(n-1)/2 terms. Row sums are the factorials (A000142). T(n,0) = T(n,1) = (n-1)! for n >= 2. T(n,2) = (n-1)!/2 = A001710(n-1) for n >= 3. Sum_{k>=0} k*T(n,k) = A067318(n).