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 A295380 Number of canonical forms for separation coordinates on hyperspheres S_n, ordered by increasing number of independent continuous parameters. 1
 1, 1, 1, 2, 3, 1, 3, 8, 5, 1, 6, 20, 22, 8, 1, 11, 49, 73, 46, 11, 1, 23, 119, 233, 206, 87, 15, 1, 46, 288, 689, 807, 485, 147, 19, 1, 98, 696, 1988, 2891, 2320, 1021, 236, 24, 1, 207, 1681, 5561, 9737, 9800, 5795, 1960, 356, 29, 1, 451, 4062, 15322, 31350, 38216, 28586, 13088, 3525, 520, 35, 1, 983, 9821, 41558, 97552, 139901, 127465, 74280, 27224, 5989, 730, 41, 1 (list; table; graph; refs; listen; history; text; internal format)
 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 Cf. A001190, A024206, A032132, A232206. Sequence in context: A100324 A121424 A214978 * A093768 A209419 A119011 Adjacent sequences:  A295377 A295378 A295379 * A295381 A295382 A295383 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

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Last modified September 18 19:15 EDT 2021. Contains 347534 sequences. (Running on oeis4.)