A209125
Triangle of coefficients of polynomials u(n,x) jointly generated with A164975; see the Formula section.
Original entry on oeis.org
1, 2, 1, 3, 4, 2, 5, 9, 9, 4, 8, 20, 25, 20, 8, 13, 40, 65, 65, 44, 16, 21, 78, 150, 190, 162, 96, 32, 34, 147, 331, 490, 521, 392, 208, 64, 55, 272, 697, 1192, 1473, 1368, 928, 448, 128, 89, 495, 1425, 2745, 3888, 4185, 3480, 2160, 960, 256, 144, 890
Offset: 1
First five rows:
1;
2, 1;
3, 4, 2;
5, 9, 9, 4;
8, 20, 25, 20, 8;
First three polynomials u(n,x):
1
2 + x
3 + 4x + 2x^2
From _Philippe Deléham_, Mar 21 2012: (Start)
(1, 1, -1, 0, 0, ...) DELTA (0, 1, 1, 0, 0, ...) begins:
1;
1, 0;
2, 1, 0;
3, 4, 2, 0;
5, 9, 9, 4, 0;
8, 20, 25, 20, 8, 0; (End)
-
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, 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[%] (* A209125 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A164975 *)
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
A164976
Triangle read by rows, expansion of 1/(1-2*y*x-x+x^2-y*x^2).
Original entry on oeis.org
1, 1, 2, 0, 5, 4, -1, 4, 16, 8, -1, -3, 25, 44, 16, 0, -10, 7, 102, 112, 32, 1, -8, -41, 97, 344, 272, 64, 1, 4, -74, -80, 528, 1040, 640, 128, 0, 15, -33, -366, 121, 2168, 2928, 1472, 256, -1, 12, 75, -426, -1219, 1898, 7664, 7840, 3328, 512
Offset: 0
Triangle begins:
1;
1, 2;
0, 5, 4;
-1, 4, 16, 8;
-1, -3, 25, 44, 16;
0, -10, 7, 102, 112, 32;
1, -8, -41, 97, 344, 272, 64;
...
-
s := 1/(1-2*y*x-x+x^2-y*x^2): t := series(s, x, 12):
seq(print(seq(coeff(coeff(t, x, n), y, m), m=0..n)), n=0..11); # Georg Fischer, Oct 15 2023
Missing commas inserted in what used to be the term "1-8-41" (now "1,-8,-41") by Geoff Bailey (geoff(AT)maths.usyd.edu.au), Nov 30 2009
Showing 1-4 of 4 results.
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