OFFSET
1,4
COMMENTS
Table 1 of the Schöbel and Veselov paper with initial 1 added. Reverse of Table 2 of the Devadoss and Read paper.
Apparently A032132 contains the row sums.
From Petros Hadjicostas, Jan 28 2018: (Start)
In this triangle, which is read by rows, for 0 <= k <= n-1 and n>=1, let T(n,k) be the number of inequivalent canonical forms for separation coordinates of the hypersphere S^n with k independent continuous parameters. It is the mirror image of sequence A232206, that is, T(n, k) = A232206(n+1, n-k) for 0 <= k <= n-1 and n>=1. (Triangular array A232206(N, K) is defined for N >= 2 and 1 <= K <= N-1.)
If B(x,y) = Sum_{n,k>=0} T(n,k)*x^n*y^k (with T(0,0) = 1, T(0,k) = 0 for k>=1, and T(n,k) = 0 for 1 <= n <= k), then B(x,y) = 1 + (x/2)*(B(x,y)^2/(1-x*y*B(x,y)) + (1 + x*y*B(x,y))*B(x^2,y^2)/(1-x^2*y^2*B(x^2,y^2))). This can be derived from the bivariate g.f. of A232206. See the comments for that sequence.
Let S(n) := Sum_{k>=0} T(n,k). The g.f. of S(n) is B(x, y=1). If we let y=1 in the above functional equation, we get x*B(x,1) = x + (1/2)*((x*B(x,1))^2/(1-x*B(x,1)) + (1 + x*B(x,1))*x^2*B(x^2,1)/(1-x^2*B(x^2,1))). After some algebra, we get 2*x*B(x,1) = x + (1/2)(x*B(x,1)/(1-x*B(x,1)) + (x*B(x,1) + x^2*B(x^2,1))/(1-x^2*B(x,1))), i.e., 2*x*B(x,1) = x + BIK(x*B(x,1)), where we have the "BIK" (reversible, indistinct, unlabeled) transform of C. G. Bower. This proves that S(n) = A032132(n+1) for n>=0, which is Copeland's claim above.
Note that for the second column we have T(n,k=2) = A048739(n-2) for 2 <= n < = 10, but T(11,2) = 4062 <> 4059 = A048739(9). In any case, they have different g.f.s (see the formula section below).
(End)
LINKS
C. G. Bower, Transforms (2)
S. Devadoss and R. C. Read, Cellular structures determined by polygons and trees, arXiv/0008145 [math.CO], 2000.
S. L. Devadoss and R. C. Read, Cellular structures determined by polygons and trees, Ann. Combin., 5 (2001), 71-98.
K. Schöbel and A. Veselov, Separation coordinates, moduli spaces, and Stasheff polytopes, arXiv:1307.6132 [math.DG], 2014.
K. Schöbel and A. Veselov, Separation coordinates, moduli spaces and Stasheff polytopes, Commun. Math. Phys., 337 (2015), 1255-1274.
FORMULA
From Petros Hadjicostas, Jan 28 2018: (Start)
G.f.: If B(x,y) = Sum_{n,k>=0} T(n,k)*x^n*y^k (with T(0,0) = 1, T(0,k) = 0 for k>=1, and T(n,k) = 0 for 1 <= n <= k), then B(x,y) = 1 + (x/2)*(B(x,y)^2/(1-x*y*B(x,y)) + (1 + x*y*B(x,y))*B(x^2,y^2)/(1-x^2*y^2*B(x^2,y^2))).
If c(N,K) = A232206(N,K) and C(x,y) = Sum_{N,K>=0} c(N,K)*x^N*y^K (with c(1,0) = 1 and c(N,K) = 0 for 0 <= N <= K), then C(x,y) = x*B(x*y, 1/y) and B(x,y) = C(x*y, 1/y)/(x*y).
Setting y=0 in the above functional equation, we get x*B(x,0) = x + (1/2)*((x*B(x,0))^2 + x^2*B(x^2,0)), which is the functional equation for the g.f. of the first column. This proves that T(n,k=0) = A001190(n+1) for n>=0 (assuming T(0,0) = 1).
The g.f. of the second column is B_1(x,0) = Sum_{n>=0} T(n,2)*x^n = lim_{y->0} (B(x,y)-B(x,0))/y, where B(x,0) = 1 + x + x^2 + ... is the g.f. of the first column. We get B_1(x,0) = x*B(x,0)*(B(x,0) - 1)/(1 - x*B(x,0)).
(End)
EXAMPLE
From Petros Hadjicostas, Jan 27 2018: (Start)
Triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9
----------------------------------------------------------------
(S^1) 1,
(S^2) 1, 1,
(S^3) 2, 3, 1,
(S^4) 3, 8, 5, 1,
(S^5) 6, 20, 22, 8, 1,
(S^6) 11, 49, 73, 46, 11, 1,
(S^7) 23, 119, 233, 206, 87, 15, 1,
(S^8) 46, 288, 689, 807, 485, 147, 19, 1,
(S^9) 98, 696, 1988, 2891, 2320, 1021, 236, 24, 1,
(S^10) 207, 1681, 5561, 9737, 9800, 5795, 1960, 356, 29, 1,
...
(End)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tom Copeland, Nov 21 2017
EXTENSIONS
Typo for T(11,3)=15322 corrected by Petros Hadjicostas, Jan 28 2018
STATUS
approved