A210558
Triangle of coefficients of polynomials v(n,x) jointly generated with A210557; see the Formula section.
Original entry on oeis.org
1, 2, 3, 3, 7, 7, 4, 12, 20, 17, 5, 18, 40, 57, 41, 6, 25, 68, 129, 158, 99, 7, 33, 105, 243, 399, 431, 239, 8, 42, 152, 410, 824, 1200, 1160, 577, 9, 52, 210, 642, 1506, 2692, 3528, 3089, 1393, 10, 63, 280, 952, 2532, 5290, 8536, 10185, 8154, 3363, 11
Offset: 1
First five rows:
1
2...3
3...7...7
4...12...20...17
5...18...40...57...41
First three polynomials v(n,x): 1, 2 + 3x , 3 + 7x + 7x^2.
-
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
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[%] (* A210557 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210558 *)
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
A164981
A triangle with Pell numbers in the first column.
Original entry on oeis.org
1, 2, 1, 5, 3, 1, 12, 10, 4, 1, 29, 30, 16, 5, 1, 70, 87, 56, 23, 6, 1, 169, 245, 185, 91, 31, 7, 1, 408, 676, 584, 334, 136, 40, 8, 1, 985, 1836, 1784, 1158, 546, 192, 50, 9, 1, 2378, 4925, 5312, 3850, 2052, 834, 260, 61, 10, 1, 5741, 13079, 15497, 12386, 7342, 3366, 1212, 341, 73, 11, 1
Offset: 1
Triangle begins
1
2,1
5,3,1
12,10,4,1
29,30,16,5,1
70,87,56,23,6,1
169,245,185,91,31,7,1
...
From _Philippe Deléham_, Oct 10 2013: (Start)
Triangle (0, 2, 1/2, -1/2, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, ...):
1
0, 1
0, 2, 1
0, 5, 3, 1
0, 12, 10, 4, 1
0, 29, 30, 16, 5, 1
0, 70, 87, 56, 23, 6, 1
0, 169, 245, 185, 91, 31, 7, 1
... (End)
-
A164981 := proc(n,k) option remember; if n <1 or k<1 or k>n then 0; elif n = 1 then 1; else 2*procname(n-1,k)+procname(n-1,k-1)+procname(n-2,k)-procname(n-2,k-1) ; end if; end proc:
-
T[n_, k_] := T[n, k] = Which[n < 1 || k < 1 || k > n, 0, n == 1, 1, True, 2*T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1]];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 06 2023 *)
-
T(n,k) = if ((n==1) && (k==1), return(1)); if ((n<=0) || (k<=0) || (nMichel Marcus, Feb 01 2023
Showing 1-3 of 3 results.
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