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A207622
Triangle of coefficients of polynomials u(n,x) jointly generated with A207623; see the Formula section.
3
1, 2, 4, 2, 7, 8, 11, 22, 4, 16, 50, 24, 22, 100, 88, 8, 29, 182, 252, 64, 37, 308, 616, 296, 16, 46, 492, 1344, 1032, 160, 56, 750, 2688, 3000, 896, 32, 67, 1100, 5016, 7656, 3696, 384, 79, 1562, 8844, 17688, 12496, 2528, 64, 92, 2158, 14872, 37752
OFFSET
1,2
COMMENTS
With offset 0, equals the stretched Riordan array ((1 - z + z^2)/(1 - z)^3, 2*z^2/(1 - z)^2) in the notation of Corsani et al., Section 2. Cf. A207616. - 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) = 2*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
4 2
7 8
11 22 4
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_] := 2 x*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[%] (* A207622 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A207623 *)
CROSSREFS
Cf. A207623, A207616, A208510, A000124 (column 1).
Sequence in context: A207612 A207620 A354766 * A335573 A073017 A296092
KEYWORD
nonn,tabf,easy
AUTHOR
Clark Kimberling, Feb 20 2012
STATUS
approved