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A209577
Triangle of coefficients of polynomials u(n,x) jointly generated with A209578; see the Formula section.
3
1, 1, 1, 3, 2, 1, 5, 6, 3, 1, 9, 13, 10, 4, 1, 15, 28, 26, 15, 5, 1, 25, 56, 64, 45, 21, 6, 1, 41, 109, 146, 124, 71, 28, 7, 1, 67, 206, 319, 315, 216, 105, 36, 8, 1, 109, 382, 671, 758, 602, 349, 148, 45, 9, 1, 177, 697, 1372, 1744, 1576, 1056, 533, 201, 55, 10
OFFSET
1,4
COMMENTS
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x) = x*u(n-1,x) + v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
The coefficients in the triangle seem to be T(n,m) = sum(k=0,n-m,2 * binomial(m+k, m)*binomial(k, n-k-m) - sum(i=0, n-m-k, binomial(m+k-1,k)*binomial(k,n-m-i-k))) by using the PARI syntax. - Thomas Baruchel, Jun 03 2018
EXAMPLE
First five rows:
1;
1, 1;
3, 2, 1;
5, 6, 3, 1;
9, 13, 10, 4, 1;
First three polynomials v(n,x): 1, 1 + x, 3 + 2x + x^2.
MATHEMATICA
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 + 1)*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[%] (* A209577 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209578 *)
CROSSREFS
Sequence in context: A132970 A192022 A208608 * A139377 A368607 A138483
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 11 2012
STATUS
approved