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A208513
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Triangle of coefficients of polynomials u(n,x) jointly generated with A111125; see the Formula section.
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6
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1, 1, 1, 1, 4, 1, 1, 9, 6, 1, 1, 16, 20, 8, 1, 1, 25, 50, 35, 10, 1, 1, 36, 105, 112, 54, 12, 1, 1, 49, 196, 294, 210, 77, 14, 1, 1, 64, 336, 672, 660, 352, 104, 16, 1, 1, 81, 540, 1386, 1782, 1287, 546, 135, 18, 1, 1, 100, 825, 2640, 4290, 4004, 2275, 800, 170
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OFFSET
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1,5
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COMMENTS
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The columns of A208513 are identical to those of A208509. Here, however, the alternating row sums are periodic (with period 1,0,-2,-3,-2,0).
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LINKS
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Table of n, a(n) for n=1..64.
Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials
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FORMULA
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u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Contribution from Peter Bala, May 01 2012: (Start)
Working with an offset of 0: T(n,0) = 1; T(n,k) = n/k*binomial(n+k-1,2*k-1) = n/k*A078812(n,k) for k > 0. Cf. A156308.
O.g.f.: ((1-t)^2+t^2*x)/((1-t)*((1-t)^2-t*x)) = 1 + (1+x)*t + (1+4*x+x^2)*t^2 + ....
u(n+1,x) = -1 + (b(2*n,x)+1)/b(n,x), where b(n,x) := sum {k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478.
This triangle is formed from the even numbered rows of A211956 with a factor of 2^(k-1) removed from the k-th column entries.
(End)
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EXAMPLE
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First five rows:
1
1...1
1...4...1
1...9...6...1
1...16...20...8...1
First five polynomials u(n,x):
1
1 + x
1 + 4x + x^2
1 + 9x + 6x^2 + x^3
1 + 16x + 20x^2 + 8x^3 + x^4
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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] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + (x + 1)*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]
Flatten[%] (* A208513 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A111125 *)
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CROSSREFS
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Cf. A111125, A208509. A078812, A085478, A156308, A211956.
Sequence in context: A208606 A136100 A183153 * A141905 A114188 A110511
Adjacent sequences: A208510 A208511 A208512 * A208514 A208515 A208516
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Feb 28 2012
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STATUS
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approved
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