A192352
Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1.
Original entry on oeis.org
1, 0, 2, 1, 9, 13, 51, 106, 322, 771, 2135, 5401, 14445, 37324, 98514, 256621, 673933, 1760997, 4615823, 12075526, 31628466, 82781215, 216761547, 567428401, 1485645049, 3889310328, 10182603746, 26657986681, 69792188337, 182717232061
Offset: 1
For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
The first six polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=1+x^2 -> 2+x
p(3,x)=3x+x^3 -> 1+5x
p(4,x)=1+6x^2+x^4 -> 9+9x
p(5,x)=5x+10x^3+x^5 -> 13+30x.
From these, we read
A192352=(1,0,2,1,9,13...) and A049602=(0,1,1,5,9,30...).
-
q[x_] := x + 1; d = 1;
p[n_, x_] := ((x + d)^n + (x - d)^n )/
2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192352 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A049602 *)
A192373
Constant term in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.
Original entry on oeis.org
1, 0, 7, 8, 77, 192, 1043, 3472, 15529, 57792, 240655, 934808, 3789653, 14963328, 60048443, 238578976, 953755537, 3798340224, 15162325975, 60438310184, 241126038941, 961476161856, 3835121918243, 15294304429744, 61000836720313, 243280700771904
Offset: 1
The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=4+x+3x^2 -> 7+4x
p(3,x)=16x+4x^2+4x^3 -> 8+28x
p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 77+84x.
From these, read A192352=(1,0,7,8,77,...) and A049602=(0,2,4,28,84,...).
-
q[x_] := x + 1; d = Sqrt[x + 4];
p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d) (* A162517 *)
Table[Expand[p[n, x]], {n, 1, 6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192373 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192374 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192375 *)
A049601
a(n)=Sum{T(2i,n-2i): i=0,1,...,[ n/2 ]}, array T as in A049600.
Original entry on oeis.org
0, 0, 2, 3, 12, 25, 76, 182, 504, 1275, 3410, 8811, 23256, 60580, 159094, 415715, 1089648, 2850645, 7466468, 19541994, 51170460, 133951675, 350713222, 918141623, 2403786672, 6293097000, 16475700746, 43133687427, 112925875764
Offset: 0
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Comments