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A207612
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Triangle of coefficients of polynomials u(n,x) jointly generated with A207613; see the Formula section.
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3
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1, 2, 4, 2, 7, 6, 4, 12, 14, 12, 8, 20, 30, 32, 24, 16, 33, 60, 76, 72, 48, 32, 54, 116, 168, 184, 160, 96, 64, 88, 218, 356, 440, 432, 352, 192, 128, 143, 402, 728, 1000, 1104, 992, 768, 384, 256, 232, 730, 1452, 2184, 2656, 2688, 2240, 1664, 768, 512
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OFFSET
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1,2
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COMMENTS
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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), v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
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EXAMPLE
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First five rows:
1
2
4....2
7....6....4
12...14...12...8
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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]
v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
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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PROG
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(Python)
from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + 2*x*v(n - 1, x) + 1
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
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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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