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A210866
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Triangle of coefficients of polynomials u(n,x) jointly generated with A210867; see the Formula section.
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4
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1, 1, 1, 1, 3, 1, 1, 6, 6, 2, 1, 10, 21, 14, 3, 1, 15, 55, 65, 31, 5, 1, 21, 120, 235, 187, 65, 8, 1, 28, 231, 700, 867, 503, 134, 13, 1, 36, 406, 1792, 3332, 2914, 1279, 268, 21, 1, 45, 666, 4074, 10955, 13882, 9084, 3122, 527, 34, 1, 55, 1035, 8430, 31563
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OFFSET
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1,5
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COMMENTS
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Row n starts with 1, followed by the n-th triangular number, and ends with the n-th Fibonacci number.
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)+x*v(n-1,x),
v(n,x)=(x+n)*u(n-1,x)+x*v(n-1,x)-x,
where u(1,x)=1, v(1,x)=1.
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EXAMPLE
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First five rows:
1
1...1
1...3....1
1...6....6....2
1...10...21...14...3
First three polynomials u(n,x): 1, 1 + x, 1 + 3x + x^2.
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x] - 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]
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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