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A209756 Triangle of coefficients of polynomials v(n,x) jointly generated with A209755; see the Formula section. 3
1, 2, 1, 3, 2, 2, 4, 4, 6, 3, 5, 7, 13, 11, 5, 6, 11, 24, 28, 22, 8, 7, 16, 40, 59, 63, 41, 13, 8, 22, 62, 110, 146, 132, 76, 21, 9, 29, 91, 188, 296, 337, 271, 138, 34, 10, 37, 128, 301, 546, 743, 754, 541, 248, 55, 11, 46, 174, 458, 938, 1477, 1793, 1632 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Column 1: 1,2,3,4,5,6,....... A000027
Column 2: 1,2,4,7,11,........ A000124
Column 3: 2,6,13,24,......... A003600
Last term in row n: 1,1,2,3,. A000045 (Fibonacci numbers)
Row sums: 1,3,7,17,41,...... A001333
Alternating row sums: 1,2,3,3,5,5,7,7,...; A109613
For a discussion and guide to related arrays, see A208510.
LINKS
FORMULA
u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
2...1
3...2...2
4...4...6....3
5...7...13...11...5
First three polynomials v(n,x): 1, 2 + x , 3 + 2x + 2x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 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[%] (* A209755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209756 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A133654 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A001333 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A109613 *)
CROSSREFS
Cf. A208510.
Sequence in context: A269456 A208906 A120933 * A210795 A210862 A298675
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 14 2012
STATUS
approved

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Last modified June 22 15:06 EDT 2024. Contains 373587 sequences. (Running on oeis4.)