A208748
Triangle of coefficients of polynomials v(n,x) jointly generated with A208747; see the Formula section.
Original entry on oeis.org
1, 0, 4, 0, 2, 12, 0, 2, 8, 40, 0, 2, 8, 40, 128, 0, 2, 8, 48, 160, 416, 0, 2, 8, 56, 208, 640, 1344, 0, 2, 8, 64, 256, 928, 2432, 4352, 0, 2, 8, 72, 304, 1248, 3840, 9088, 14080, 0, 2, 8, 80, 352, 1600, 5504, 15616, 33280, 45568, 0, 2, 8, 88, 400, 1984, 7424
Offset: 1
First five rows:
1
0...4
0...2...12
0...2...8...40
0...2...8...40...128
First five polynomials v(n,x):
1
4x
2x + 12x^2
2x + 8x^2 + 40x^3
2x + 8x^2 + 40x^3 + 128x^4
-
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x];
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[%] (* A208747 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208748 *)
A208510
Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 9, 5, 1, 1, 9, 16, 14, 6, 1, 1, 11, 25, 30, 20, 7, 1, 1, 13, 36, 55, 50, 27, 8, 1, 1, 15, 49, 91, 105, 77, 35, 9, 1, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 1, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1
Offset: 1
First five rows:
1
1...1
1...3...1
1...5...4...1
1...7...9...5...1
First five polynomials u(n,x):
1
1 + x
1 + 3x + x^2
1 + 5x + 4x^2 + x^3
1 + 7x + 9x^2 + 5x^3 + x^4
-
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_] := u[n - 1, x] + 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[%] (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A029653 *)
-
from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017
A208342
Triangle of coefficients of polynomials u(n,x) jointly generated with A208343; see the Formula section.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 10, 8, 1, 1, 6, 9, 16, 18, 13, 1, 1, 7, 11, 23, 31, 33, 21, 1, 1, 8, 13, 31, 47, 62, 59, 34, 1, 1, 9, 15, 40, 66, 101, 119, 105, 55, 1, 1, 10, 17, 50, 88, 151, 205, 227, 185, 89, 1, 1, 11, 19, 61, 113, 213, 321, 414
Offset: 1
First five rows:
1
1, 1
1, 1, 2
1, 1, 3, 3
1, 1, 4, 5, 5
First five polynomials u(n,x): 1, 1 + x, 1 + x + x^2, 1 + x + 3*x^2 + 3*x^3, 1 + x + 4*x^2 + 5*x^3 + 5*x^4.
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
1
1, 0
1, 1, 0
1, 1, 2, 0
1, 1, 3, 3, 0
1, 1, 4, 5, 5, 0
1, 1, 5, 7, 10, 8, 0
1, 1, 6, 9, 16, 18, 13, 0
1, 1, 7, 11, 23, 31, 33, 21, 0
-
u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
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[%] (* A208342 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208343 *)
Showing 1-3 of 3 results.
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