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A172431
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Even row Pascal-square read by anti-diagonals.
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6
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1, 1, 2, 1, 4, 3, 1, 6, 10, 4, 1, 8, 21, 20, 5, 1, 10, 36, 56, 35, 6, 1, 12, 55, 120, 126, 56, 7, 1, 14, 78, 220, 330, 252, 84, 8, 1, 16, 105, 364, 715, 792, 462, 120, 9, 1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10
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OFFSET
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1,3
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COMMENTS
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Apart from signs identical to A053123. Mirror of A078812.
As a triangle, row n consists of the coefficients of Morgan-Voyce polynomial B(n,x); e.g., B(3,x)=x^3+6x^2+10x+4. As a triangle, rows 0 to 4 are as follows: 1 1...2 1...4...3 1...6...10...4 1...8...21...20...5 See A054142 for coefficients of Morgan-Voyce polynomial b(n,x).
Scaled version of A119900. - Philippe Deléham, Feb 24 2012
A172431 is jointly generated with A054142 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+v(n-1)x and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section. [From Clark Kimberling, Mar 9 2012]
Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 22 2012
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LINKS
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Table of n, a(n) for n=1..55.
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FORMULA
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As a decimal sequence: a(n)= 12*a(n-1)- a(n-2) with a(1)=1.
As triangle T(n,k) : T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2). - Philippe Deléham, Feb 24 2012
As DELTA-triangle T(n,k) with 0<=k<=n : G.f.: (1-y*x)^2/((1-y*x)^2-x).- Philippe Deléham, Mar 22 2012
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EXAMPLE
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Array begins:
1,2,3,4,5,...
1,4,10,20,...
1,6,21,56,...
Example:
Starting with 1, every entry is twice the one to the left minus the second one to the left, plus the one above.
For n = 9 the a(9)= 10 solution is 2*4 - 1 + 3
Triangle T(n,k) begins :
1
1, 2
1, 4, 3
1, 6, 10, 4
1, 8, 21, 20, 5
1, 10, 36, 56, 35, 6
1, 12, 55, 120, 126, 56, 7 - Philippe Deléham, Feb 24 2012
(1, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, 1/2, 0, 0, ...) begins :
1
1, 0
1, 2, 0
1, 4, 3, 0
1, 6, 10, 4, 0
1, 8, 21, 20, 5, 0
1, 10, 36, 56, 35, 6, 0
1, 12, 55, 120, 126, 56, 7, 0 . - Philippe Deléham, Mar 22 2012
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := 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]
Flatten[%] (* A054142 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A172431 *)
(* Clark Kimberling, Mar 9 2012 *)
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CROSSREFS
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Cf. A078812, A053123, A007318, A001906
Cf. A054142, A082985
Sequence in context: A093190 A132191 A094437 * A053123 A107661 A126570
Adjacent sequences: A172428 A172429 A172430 * A172432 A172433 A172434
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KEYWORD
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nonn,tabl
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AUTHOR
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M. Dols (markdols99(AT)yahoo.com), Feb 02 2010
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EXTENSIONS
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Added a comment. - Clark Kimberling, Feb 13 2010
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STATUS
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approved
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