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
A209415
Triangle of coefficients of polynomials u(n,x) jointly generated with A209416; see the Formula section.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 4, 6, 1, 1, 6, 11, 10, 1, 1, 7, 21, 25, 15, 1, 1, 9, 30, 57, 50, 21, 1, 1, 10, 45, 99, 133, 91, 28, 1, 1, 12, 58, 168, 275, 280, 154, 36, 1, 1, 13, 78, 250, 523, 675, 546, 246, 45, 1, 1, 15, 95, 370, 885, 1433, 1509, 1002, 375, 55, 1, 1, 16, 120, 505, 1435, 2718, 3564, 3135, 1749, 550, 66, 1
Offset: 1
First five rows:
1;
1, 1;
1, 3, 1;
1, 4, 6, 1;
1, 6, 11, 10, 1;
First three polynomials v(n,x): 1, 1 + x, 1 + 3x + x^2.
From _Philippe Deléham_, Apr 02 2012: (Start)
(1, 0, 1, -2, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, ...) begins:
1;
1, 0;
1, 1, 0;
1, 3, 1, 0;
1, 4, 6, 1, 0;
1, 6, 11, 10, 1, 0;
1, 7, 21, 25, 15, 1, 0;
1, 9, 30, 57, 50, 21, 1, 0;
1, 10, 45, 99, 133, 91, 28, 1, 0; (End)
-
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := (x + 1)*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[%] (* A209415 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209416 *)
CoefficientList[CoefficientList[Series[(1 + x - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2), {x,0,10}, {y,0,10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)
A208335
Triangle of coefficients of polynomials v(n,x) jointly generated with A208834; see the Formula section.
Original entry on oeis.org
1, 2, 1, 3, 3, 1, 4, 7, 5, 1, 5, 14, 15, 6, 1, 6, 25, 36, 23, 8, 1, 7, 41, 76, 69, 36, 9, 1, 8, 63, 147, 176, 123, 48, 11, 1, 9, 92, 266, 400, 355, 192, 66, 12, 1, 10, 129, 456, 834, 910, 635, 292, 82, 14, 1, 11, 175, 747, 1626, 2131, 1833, 1065, 410, 105, 15, 1
Offset: 1
First five rows:
1;
2, 1;
3, 3, 1;
4, 7, 5, 1;
5, 14, 15, 6, 1;
First five polynomials v(n,x):
1
2 + x
3 + 3x + x^2
4 + 7x + 5x^2 + x^3
5 + 14x + 15x^2 + 6x^3 + x^4
From _Philippe Deléham_, Mar 26 2012: (Start)
(1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, ...) begins:
1;
1, 0;
2, 1, 0;
3, 3, 1, 0;
4, 7, 5, 1, 0;
5, 14, 15, 6, 1, 0; (End)
-
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 + 1)*u[n - 1, 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[%] (* A208334 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208335 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* u row sums *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* v row sums *)
Table[u[n, x] /. x -> -1, {n, 1, z}](* u alt. row sums *)
Table[v[n, x] /. x -> -1, {n, 1, z}](* v alt. row sums *)
Showing 1-3 of 3 results.
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