A164981 -id:A164981 - cp's OEIS frontend

A210557 Triangle of coefficients of polynomials u(n,x) jointly generated with A210558; see the Formula section.

1, 1, 2, 1, 3, 5, 1, 4, 10, 12, 1, 5, 16, 30, 29, 1, 6, 23, 56, 87, 70, 1, 7, 31, 91, 185, 245, 169, 1, 8, 40, 136, 334, 584, 676, 408, 1, 9, 50, 192, 546, 1158, 1784, 1836, 985, 1, 10, 61, 260, 834, 2052, 3850, 5312, 4925, 2378, 1, 11, 73, 341, 1212, 3366

Offset: 1

Clark Kimberling, Mar 22 2012

Row sums: powers of 3 (see A000244).
For a discussion and guide to related arrays, see A208510.
Subtriangle of (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 23 2012
Up to reflection at the vertical axis, this triangle coincides with the triangle given in A164981, i.e., the numbers are the same just read row-wise in the opposite direction. - Christine Bessenrodt, Jul 20 2012

			First five rows:
  1;
  1, 2;
  1, 3,  5;
  1, 4, 10, 12;
  1, 5, 16, 30, 29;
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 5x^2.
From _Philippe Deléham_, Mar 23 2012: (Start)
(1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, ...) begins:
  1;
  1, 0;
  1, 2,  0;
  1, 3,  5,  0;
  1, 4, 10, 12,  0;
  1, 5, 16, 30, 29, 0; (End)

Cf. A210558, A208510, A164981.

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 *)

u(n,x) = x*u(n-1,x) + x*v(n-1,x)+1,
v(n,x) = 2x*u(n-1,x) + (x+1)v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 23 2012. (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1 - 2*y*x + y*x^2 - y^2*x^2)/(1 - x - 2*y*x + y*x^2 - y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)

A164984 Odd (Jacobsthal) triangle.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 5, 9, 5, 1, 11, 23, 19, 7, 1, 21, 57, 61, 33, 9, 1, 43, 135, 179, 127, 51, 11, 1, 85, 313, 493, 433, 229, 73, 13, 1, 171, 711, 1299, 1359, 891, 375, 99, 15, 1, 341, 1593, 3309, 4017, 3141, 1641, 573, 129, 17, 1

Offset: 1

Mark Dols, Sep 03 2009, Sep 06 2009

Alternate diagonal sums give A008619.

Diagonals sums give A097076. - Philippe Deléham, Oct 13 2013

			1
1,1
3,3,1
5,9,5,1
11,23,19,7,1
21,57,61,33,9,1
Pascal-like triangle based on a right-triangular sum (with the top multiplied by 2): For n=13 a(13)=2*a(3)+a(5)+a(8)+a(9)= 2+3+9+5=19.

Cf. A164981, A001045, A005408, A015518 (row sums), A008619

Cf. A008288, A007318.

Excel formula: C6=2*C4+C5+B5+B4 with C5=a(1)=1 and C6=a(2)

T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k) + T(n-2,k-1). - Philippe Deléham, Oct 13 2013

cp's OEIS Frontend

A210557 Triangle of coefficients of polynomials u(n,x) jointly generated with A210558; see the Formula section.

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