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A208509 Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section. 4
1, 3, 5, 1, 7, 5, 9, 14, 1, 11, 30, 7, 13, 55, 27, 1, 15, 91, 77, 9, 17, 140, 182, 44, 1, 19, 204, 378, 156, 11, 21, 285, 714, 450, 65, 1, 23, 385, 1254, 1122, 275, 13, 25, 506, 2079, 2508, 935, 90, 1, 27, 650, 3289, 5148, 2717, 442, 15, 29, 819, 5005, 9867 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
FORMULA
u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x) + 1, with u(1,x)=1, v(1,x)=1.
Conjecture: T(n,k) = binomial(n-1,2*k+1) + binomial(n,2*k+1). - Knud Werner, Jan 11 2022
EXAMPLE
First five rows:
1
3
5 1
7 5
9 14 1
First five polynomials v(n,x):
1
3
5 + x
7 + 5x
9 + 14x + x^2
MATHEMATICA
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] + 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[%] (* A208508 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208509 *)
CROSSREFS
Row sums, v(n,1): A003948.
Alternating row sums, v(n,-1): A090131.
Cf. A208508.
Sequence in context: A152204 A114216 A337128 * A086233 A353413 A318190
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Feb 27 2012
STATUS
approved

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Last modified February 28 21:38 EST 2024. Contains 370400 sequences. (Running on oeis4.)