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A210040
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Array of coefficients of polynomials v(n,x) jointly generated with A210039; see the Formula section.
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3
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1, 2, 1, 3, 4, 4, 10, 1, 5, 20, 6, 6, 35, 21, 1, 7, 56, 56, 8, 8, 84, 126, 36, 1, 9, 120, 252, 120, 10, 10, 165, 462, 330, 55, 1, 11, 220, 792, 792, 220, 12, 12, 286, 1287, 1716, 715, 78, 1, 13, 364, 2002, 3432, 2002, 364, 14, 14, 455, 3003, 6435, 5005, 1365
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OFFSET
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1,2
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COMMENTS
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Every term is a binomial coefficient.
For a discussion and guide to related arrays, see A208510.
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LINKS
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FORMULA
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u(n,x)=u(n-1,x)+v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Also: writing T(n,m) for the general term,
T(n,1)=n for n>=1;
T(n,k)=C(n+1,2k-1) for 1<=k<=floor[(n+2)/2].
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EXAMPLE
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First eight rows:
1
2...1
3...4
4...10...1
5...20...6
6...35...21....1
7...56...56....8
8...84...126...36...1
First five polynomials v(n,x):
1
2 + x
3 + 4x.
4 + 10x + x^2
5 + 20x + 6x^2.
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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