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A208525
Triangle of coefficients of polynomials v(n,x) jointly generated with A208524; see the Formula section.
3
1, 2, 3, 3, 7, 5, 4, 12, 18, 11, 5, 18, 42, 49, 21, 6, 25, 80, 135, 116, 43, 7, 33, 135, 295, 381, 279, 85, 8, 42, 210, 560, 966, 1050, 638, 171, 9, 52, 308, 966, 2086, 2996, 2724, 1453, 341, 10, 63, 432, 1554, 4032, 7182, 8688, 6921, 3240, 683, 11, 75
OFFSET
1,2
COMMENTS
Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...
FORMULA
u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
2...3
3...7....5
4...12...18...11
5...18...42...49...21
First five polynomials v(n,x):
1
2 + 3x
3 + 7x + 5x^2
4 + 12x + 18x^2 + 11x^3
5 + 18x + 42x^2 + 49x^3 + 21x^4
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_] := 2 x*u[n - 1, x] + (x + 1)*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[%] (* A208524 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208525 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (*A060816*)
Table[v[n, x] /. x -> 1, {n, 1, z}] (*|A084244|*)
Table[u[n, x] /. x -> -1, {n, 1, z}] (*alt. row sums*)
Table[v[n, x] /. x -> -1, {n, 1, z}] (*alt. row sums*)
CROSSREFS
Cf. A208524.
Sequence in context: A088100 A097359 A358182 * A342878 A209574 A079387
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 29 2012
STATUS
approved