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A109001
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Triangle, read by rows, where g.f. of row n equals the product of (1-x)^n and the g.f. of the coordination sequence for root lattice B_n, for n>=0.
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3
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1, 1, 1, 1, 6, 1, 1, 15, 23, 1, 1, 28, 102, 60, 1, 1, 45, 290, 402, 125, 1, 1, 66, 655, 1596, 1167, 226, 1, 1, 91, 1281, 4795, 6155, 2793, 371, 1, 1, 120, 2268, 12040, 23750, 18888, 5852, 568, 1, 1, 153, 3732, 26628, 74574, 91118, 49380, 11124, 825, 1, 1, 190, 5805
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Compare to triangle A108558, where row n equals the (n+1)-th differences of the crystal ball sequence for D_n lattice.
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REFERENCES
| R. Bacher, P. de la Harpe and B. Venkov, Series de croissance et series d'Ehrhart associees aux reseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
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FORMULA
| T(n, k) = C(2*n+1, 2*k) - 2*n*C(n-1, k-1). Row sums are: 2^n*(2^n - n) for n>=0. G.f. for coordination sequence of B_n lattice: Sum(binomial(2*n+1, 2*i)*z^i, i=0..n)-2*n*z*(1+z)^(n-1))/(1-z)^n. [Bacher et al.]
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EXAMPLE
| G.f.s of initial rows of square array A108998 are:
(1),
(1 + x)/(1-x),
(1 + 6*x + x^2)/(1-x)^2;
(1 + 15*x + 23*x^2 + x^3)/(1-x)^3;
(1 + 28*x + 102*x^2 + 60*x^3 + x^4)/(1-x)^4.
Triangle begins:
1;
1,1;
1,6,1;
1,15,23,1;
1,28,102,60,1;
1,45,290,402,125,1;
1,66,655,1596,1167,226,1;
1,91,1281,4795,6155,2793,371,1;
1,120,2268,12040,23750,18888,5852,568,1;
1,153,3732,26628,74574,91118,49380,11124,825,1; ...
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PROG
| (PARI) T(n, k)=binomial(2*n+1, 2*k)-2*n*binomial(n-1, k-1)
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CROSSREFS
| Cf. A108998, A108999, A109000, A022144 (row 2), A022145 (row 3), A022146 (row 4), A022147 (row 5), A022148 (row 6), A022149 (row 7), A022150 (row 8), A022151 (row 9), A022152 (row 10), A022153 (row 11), A022154 (row 12).
Sequence in context: A176152 A146958 A154653 * A203005 A176560 A152602
Adjacent sequences: A108998 A108999 A109000 * A109002 A109003 A109004
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jun 17 2005
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