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A207629
Triangle of coefficients of polynomials u(n,x) jointly generated with A207630; see the Formula section.
4
1, 2, 5, 1, 11, 4, 23, 13, 1, 47, 37, 6, 95, 97, 25, 1, 191, 241, 87, 8, 383, 577, 271, 41, 1, 767, 1345, 783, 169, 10, 1535, 3073, 2143, 609, 61, 1, 3071, 6913, 5631, 2001, 291, 12, 6143, 15361, 14335, 6145, 1191, 85, 1, 12287, 33793, 35583, 17921
OFFSET
1,2
COMMENTS
With offset 0, equals the stretched Riordan array ((1 - z + z^2)/(1 - 3*z + 2*z^2), z^2/(1 - 2*z)) in the notation of Corsani et al., Section 2. - Peter Bala, Dec 31 2015
LINKS
C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.
FORMULA
u(n,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.
EXAMPLE
First five rows:
1
2
5 1
11 4
23 13 1
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := (x + 1)*u[n - 1, 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]
Flatten[%] (* A207629 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A207630 *)
CROSSREFS
Cf. A207630, A208510, A083329 (column 1).
Sequence in context: A120235 A323411 A089618 * A207614 A156067 A352577
KEYWORD
nonn,tabf,easy
AUTHOR
Clark Kimberling, Feb 23 2012
STATUS
approved