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A208330
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Triangle of coefficients of polynomials u(n,x) jointly generated with A208331; see the Formula section.
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3
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1, 1, 1, 1, 2, 3, 1, 3, 9, 5, 1, 4, 18, 20, 11, 1, 5, 30, 50, 55, 21, 1, 6, 45, 100, 165, 126, 43, 1, 7, 63, 175, 385, 441, 301, 85, 1, 8, 84, 280, 770, 1176, 1204, 680, 171, 1, 9, 108, 420, 1386, 2646, 3612, 3060, 1539, 341, 1, 10, 135, 600, 2310, 5292, 9030
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OFFSET
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1,5
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COMMENTS
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Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012
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LINKS
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FORMULA
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u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 18 2012
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EXAMPLE
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First five rows:
1
1...1
1...2...3
1...3...9....5
1...4...18...20...11
First five polynomials u(n,x):
1, 1 + x, 1 + 2x + 3x^2, 1 + 3x + 9x^2 + 5x^3, 1 + 4x + 18x^2 + 20x^3 + 11x^4.
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, ...) begins :
1
1, 0
1, 1, 0
1, 2, 3, 0
1, 3, 9, 5, 0
1, 4, 18, 20, 11, 0
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
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]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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