A208510 -id:A208510 - cp's OEIS frontend

A210875 Triangular array U(n,k) of coefficients of polynomials defined in Comments.

1, 1, 1, 3, 4, 2, 4, 7, 5, 3, 5, 9, 10, 9, 5, 6, 11, 13, 17, 14, 8, 7, 13, 16, 22, 27, 23, 13, 8, 15, 19, 27, 35, 44, 37, 21, 9, 17, 22, 32, 43, 57, 71, 60, 34, 10, 19, 25, 37, 51, 70, 92, 115, 97, 55, 11, 21, 28, 42, 59, 83, 113, 149, 186, 157, 89, 12, 23, 31, 47, 67

Offset: 1

Clark Kimberling, Mar 30 2012

Polynomials u(n,k) are defined by u(n,x)=x*u(n-1,x)+(x^2)*u(n-2,x)+n*(x+1), where u(1)=1 and u(2,x)=x+1.  The array (U(n,k)) is defined by rows:
u(n,x)=U(n,1)+U(n,2)*x+...+U(n,n-1)*x^(n-1).
In each column, the first number is a Fibonacci number and, with one exception, the difference between each two consecutive terms is a Fibonacci number (see the Formula section).
Alternating row sums: 1,0,1,-2,3,-5,8,-13,21,... (signed Fibonacci numbers)

			First six rows:
  1
  1...1
  3...4....2
  4...7....5....3
  5...9....10...9....5
  6...11...13...17...14...8
First three polynomials u(n,x): 1, 1 + 3x, 3 + 4x + 2x^2.

Cf. A208510, A210881, A210874.

u[1, x_] := 1; u[2, x_] := x + 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + (x^2)*u[n - 2, x] + n*(x + 1);
Table[Expand[u[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210875 *)

Column k consists of the partial sums of the following sequence: F(k), 3*F(k-1), F(k+2), F(k+1), F(k+1),..., where F=A000045 (Fibonacci numbers). That is, U(n+1,k)-U(n,k)=F(k+1) for n>2.

A210881 Triangular array U(n,k) of coefficients of polynomials defined in Comments.

Original entry on oeis.org

1, 1, 3, 3, 4, 4, 4, 7, 5, 7, 5, 9, 10, 9, 11, 6, 11, 13, 17, 14, 18, 7, 13, 16, 22, 27, 23, 29, 8, 15, 19, 27, 35, 44, 37, 47, 9, 17, 22, 32, 43, 57, 71, 60, 76, 10, 19, 25, 37, 51, 70, 92, 115, 97, 123, 11, 21, 28, 42, 59, 83, 113, 149, 186, 157, 199, 12, 23, 31

Offset: 1

Clark Kimberling, Mar 30 2012

Polynomials u(n,k) are defined by u(n,x)=x*u(n-1,x)+(x^2)*u(n-2,x)+n*(x+1), where u(1)=1 and u(2,x)=3x+1.  The array (U(n,k)) is defined by rows:
u(n,x)=U(n,1)+U(n,2)*x+...+U(n,n-1)*x^(n-1).
In each column, the first number is a Lucas number and the difference between each two consecutive terms is a Fibonacci number (see the Formula section).
Alternating row sums: 1,-2,3,-5,8,-13,21,... (signed Fibonacci numbers)

			First six rows:
1
1...3
3...4....4
4...7....5....7
5...9....10...9....11
6...11...13...17...14...18
First three polynomials u(n,x): 1, 1 + 3x, 3 + 4x + 4x^2.

Cf. A208510, A210874, A210875.

u[1, x_] := 1; u[2, x_] := 3 x + 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + (x^2)*u[n - 2, x] + n*(x + 1);
Table[Expand[u[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210881 *)

Column k consists of the partial sums of the following sequence: L(k), F(k-1), F(k+2), F(k+1), F(k+1), F(k+1),..., where L=A000032 (Lucas numbers) and F=000045 (Fibonacci numbers). That is, U(n+1,k)-U(n,k)=F(k+1) for n>2.

A131084 A129686 * A007318. Riordan triangle (1+x, x/(1-x)).

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 2, 3, 1, 0, 2, 5, 4, 1, 0, 2, 7, 9, 5, 1, 0, 2, 9, 16, 14, 6, 1, 0, 2, 11, 25, 30, 20, 7, 1, 0, 2, 13, 36, 55, 50, 27, 8, 1, 0, 2, 15, 49, 91, 105, 77, 35, 9, 1

Offset: 1

Gary W. Adamson, Jun 14 2007

Row sums = A098011 starting (1, 2, 3, 6, 12, 24, 48, ...). A131085 = A007318 * A129686

Riordan array (1+x, x/(1-x)). - Philippe Deléham, Mar 02 2012

			The triangle T(n, k) begins:
n\k 0  1  2  3   4   5   6   7  8  9 10 ...
0:  1
1:  1  1
2:  0  2  1
3:  0  2  3  1
4:  0  2  5  4   1
5:  0  2  7  9   5   1
6:  0  2  9 16  14   6   1
7:  0  2 11 25  30  20   7   1
8:  0  2 13 36  55  50  27   8  1
9:  0  2 15 49  91 105  77  35  9  1
10: 0  2 17 64 140 196 182 112 44 10  1
... Reformatted. - _Wolfdieter Lang_, Jan 06 2015

Cf. A007318, A129686, A098011, A131085.

Cf. A029653, A131084, A208510.

A129686(signed): (1,1,1,...) in the main diagonal and (-1,-1,-1, ...) in the subsubdiagonal); * A007318, Pascal's triangle; as infinite lower triangular matrices.
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(2*x + 3*x^2/2! + x^3/3!) = 2*x + 7*x^2/2! + 16*x^3/3! + 30*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014
G.f. column k: (1+x)*(x/(1-x))^k, k >= 0. (Riordan property). - Wolfdieter Lang, Jan 06 2015
T(n, 0) = 1 if n=0 or n=1 else 0; T(n, k) = binomial(n-1,k-1) + binomial(n-2,k-1)*[n-1 >= k] if n >= k >= 1, where [S] = 1 if S is true, else 0, and  T(n, k) = 0 if n < k. - Wolfdieter Lang, Jan 08 2015

Edited: Added Riordan property (see Philippe Deléham comment) in name. - Wolfdieter Lang, Jan 06 2015

A185045 Triangle of coefficients of polynomials u(n,x) jointly generated with A208659; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 6, 4, 1, 10, 16, 8, 1, 14, 36, 40, 16, 1, 18, 64, 112, 96, 32, 1, 22, 100, 240, 320, 224, 64, 1, 26, 144, 440, 800, 864, 512, 128, 1, 30, 196, 728, 1680, 2464, 2240, 1152, 256, 1, 34, 256, 1120, 3136, 5824, 7168, 5632, 2560, 512, 1, 38, 324

Offset: 1

Clark Kimberling, Mar 03 2012

Alternating row sums: 1,-1,-1,-1,-1,-1,-1,-1,-1,...

For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
1...6...4
1...10...16...8
1...14...36...40...16
First five polynomials u(n,x):
1
1 + 2x
1 + 6x + 4x^2
1 + 10x + 16x^2 + 8x^3
1 + 14x + 36x^2 + 40x^3 + 16x^4

Cf. A208659, A208510.

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_] := u[n - 1, x] + 2 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[%]    (* A185045 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208659 *)

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1), T(1,0) = T(2,0) = T(3,0) = 1, T(2,1) = 2, T(3,1) = 6, T(3,2) = 4, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 19 2012

A207611 Triangle of coefficients of polynomials v(n,x) jointly generated with A207610; see Formula section.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 8, 8, 5, 2, 1, 13, 15, 11, 6, 2, 1, 21, 28, 23, 14, 7, 2, 1, 34, 51, 47, 32, 17, 8, 2, 1, 55, 92, 93, 70, 42, 20, 9, 2, 1, 89, 164, 181, 148, 97, 53, 23, 10, 2, 1, 144, 290, 346, 306, 217, 128, 65, 26, 11, 2, 1, 233, 509, 653, 619, 472

Offset: 1

Clark Kimberling, Feb 19 2012

Column 1:  Fibonacci numbers, A000045
Column 2:  A029907
Row sums:  A003945.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

			First five rows:
  1;
  2, 1;
  3, 2, 1;
  5, 4, 2, 1;
  8, 8, 5, 2, 1;
From _Philippe Deléham_, Mar 25 2012: (Start)
(0, 2, -1/2, -1/2, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, ...) begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  2,  1;
  0,  5,  4,  2,  1;
  0,  8,  8,  5,  2,  1;
  0, 13, 15, 11,  6,  2,  1;
  0, 21, 28, 23, 14,  7,  2,  1; (End)

Cf. A207610, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207610 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207611 *)
T[ n_, k_] := Which[k<0 || n<0, 0, n<2, Boole[k<=n] + Boole[k==0&&n==1], True, T[n, k] = T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1] ]; (* Michael Somos, Sep 19 2024 *)

{T(n, k) = if(k<0 || n<0, 0, n<2, (k<=n) + (k==0 && n==1), T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-2, k-1) )}; /* Michael Somos, Sep 19 2024 */

from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, 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(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = u(n-1,x) + x*v(n-1,x)+1, where u(1,x)=1, v(1,x)=1.

T(n,k) = T(n-1,k) + (n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(1,0) = T(2,1) = 1, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k >= n.

A207613 Triangle of coefficients of polynomials v(n,x) jointly generated with A207612; see Formula section.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 8, 8, 8, 8, 16, 20, 16, 16, 13, 30, 44, 48, 32, 32, 21, 56, 92, 112, 112, 64, 64, 34, 102, 188, 256, 272, 256, 128, 128, 55, 184, 372, 560, 672, 640, 576, 256, 256, 89, 328, 724, 1184, 1552, 1696, 1472, 1280, 512, 512, 144, 580, 1384

Offset: 1

Clark Kimberling, Feb 19 2012

Only column 1 contains odd numbers.
column 1:  A000045 (Fibonacci sequence)
row sums:  A002878 (bisection of Lucas sequence)
top edge:  A000079 (powers of 2)

			First five rows:
  1
  2  2
  3  4  4
  5  8  8  8
  8 16 20 16 16

A000045 (column 1), A000079 (main diagonal), A002878 (row sums). Cf. A207612, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207612 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207613 *)

u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = u(n-1,x) + 2x*v(n-1,x) + 1, where u(1,x) = 1, v(1,x) = 1.

With offset 0, the Riordan array ((1 + z)/(1 - z - z^2), 2*z*(1 - z)/(1 - z - z^2)) with o.g.f. (1 + z)/(1 - z - z^2 - x*(2*z - 2*z^2)) = 1 + (2 + 2*x)*z + (3 + 4*x + 4*x^2)*z^2 + .... - Peter Bala, Dec 30 2015

A207622 Triangle of coefficients of polynomials u(n,x) jointly generated with A207623; see the Formula section.

Original entry on oeis.org

1, 2, 4, 2, 7, 8, 11, 22, 4, 16, 50, 24, 22, 100, 88, 8, 29, 182, 252, 64, 37, 308, 616, 296, 16, 46, 492, 1344, 1032, 160, 56, 750, 2688, 3000, 896, 32, 67, 1100, 5016, 7656, 3696, 384, 79, 1562, 8844, 17688, 12496, 2528, 64, 92, 2158, 14872, 37752

Offset: 1

Clark Kimberling, Feb 20 2012

With offset 0, equals the stretched Riordan array ((1 - z + z^2)/(1 - z)^3, 2*z^2/(1 - z)^2) in the notation of Corsani et al., Section 2. Cf. A207616. - Peter Bala, Dec 31 2015

			First five rows:
   1
   2
   4  2
   7  8
  11 22  4

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Cf. A207623, A207616, A208510, A000124 (column 1).

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207622 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207623 *)

u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = 2*x*u(n-1,x) + v(n-1,x) + 1, where u(1,x) = 1, v(1,x) = 1.

A208330 Triangle of coefficients of polynomials u(n,x) jointly generated with A208331; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 9, 5, 1, 4, 18, 20, 11, 1, 5, 30, 50, 55, 21, 1, 6, 45, 100, 165, 126, 43, 1, 7, 63, 175, 385, 441, 301, 85, 1, 8, 84, 280, 770, 1176, 1204, 680, 171, 1, 9, 108, 420, 1386, 2646, 3612, 3060, 1539, 341, 1, 10, 135, 600, 2310, 5292, 9030

Offset: 1

Clark Kimberling, Feb 26 2012

Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

			First five rows:
1
1...1
1...2...3
1...3...9....5
1...4...18...20...11
First five polynomials u(n,x):
1, 1 + x, 1 + 2x + 3x^2, 1 + 3x + 9x^2 + 5x^3, 1 + 4x + 18x^2 + 20x^3 + 11x^4.
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, ...) begins :
1
1, 0
1, 1, 0
1, 2, 3, 0
1, 3, 9, 5, 0
1, 4, 18, 20, 11, 0
1, 5, 30, 50, 55, 21, 0. - _Philippe Deléham_, Mar 18 2012

Cf. A208331, A208510.

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_] := 2 x*u[n - 1, x] + (x + 1)*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[%]  (* A208330 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208331 *)

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = A001045(k+1)*binomial(n-1,k). - Philippe Deléham, Mar 18 2012
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 18 2012

A208344 Triangle of coefficients of polynomials u(n,x) jointly generated with A208345; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 5, 10, 17, 1, 1, 6, 13, 27, 41, 1, 1, 7, 16, 38, 71, 99, 1, 1, 8, 19, 50, 106, 186, 239, 1, 1, 9, 22, 63, 146, 294, 484, 577, 1, 1, 10, 25, 77, 191, 424, 806, 1253, 1393, 1, 1, 11, 28, 92, 241, 577, 1212, 2191, 3229, 3363, 1, 1, 12

Offset: 1

Clark Kimberling, Feb 25 2012

Row sums, u(n,1):  (1,2,5,13,...), odd-indexed Fibonacci numbers.
Row sums, v(n,1):  (1,3,8,21,...), even-indexed Fibonacci numbers.
Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 09 2012

			First five rows:
  1;
  1,  1;
  1,  1,  3;
  1,  1,  4,  7;
  1,  1,  5, 10, 17;
First five polynomials u(n,x):
  1
  1 + x
  1 + x + 3x^2
  1 + x + 4x^2 + 7x^3
  1 + x + 5x^2 + 10x^3 + 17x^4.
From _Philippe Deléham_, Apr 09 2012: (Start)
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins:
  1;
  1,  0;
  1,  1,  0;
  1,  1,  3,  0;
  1,  1,  4,  7,  0;
  1,  1,  5, 10, 17,  0;
  1,  1,  6, 13, 27, 41,  0; (End)

Cf. A208345, A208510.

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] + 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[%]    (* A208344 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208345 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]
Table[v[n, x] /. x -> 1, {n, 1, z}]

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Apr 09 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+2*y*x^2-y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) -2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
Working with an offset of 0, the row reversed triangle is the Riordan array ( (1 - x)/(1 - 2*x - x^2), x*(1 - 2*x)/(1 - 2*x - x^2) ) with g.f. (1 - x)/(1 - (2 + y)*x - (1 - 2*y)*x^2) = 1 + (1 + y)*x + (3 + y + y^2)*x^2 + (7 + 4*y + y^2 + y^3)*x^3 + .... - Peter Bala, Jun 01 2024

a(69) corrected by Georg Fischer, Sep 03 2021

A208659 Triangle of coefficients of polynomials v(n,x) jointly generated with A185045; see the Formula section.

Original entry on oeis.org

1, 2, 2, 2, 6, 4, 2, 10, 16, 8, 2, 14, 36, 40, 16, 2, 18, 64, 112, 96, 32, 2, 22, 100, 240, 320, 224, 64, 2, 26, 144, 440, 800, 864, 512, 128, 2, 30, 196, 728, 1680, 2464, 2240, 1152, 256, 2, 34, 256, 1120, 3136, 5824, 7168, 5632, 2560, 512, 2, 38, 324

Offset: 1

Clark Kimberling, Mar 03 2012

Alternating row sums:  1, 0, 0, 0, 0, 0, 0, 0, 0, ...
For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0 <= k <= n, it is (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012

			First five rows:
  1;
  2,  2;
  2,  6,  4;
  2, 10, 16,  8;
  2, 14, 36, 40, 16;
First five polynomials v(n,x):
  1
  2 +  2x = 2*(1+x)
  2 +  6x +  4x^2 = 2*(1+x)*(1+2x)
  2 + 10x + 16x^2 +  8x^3 = 2*(1+x)*(1+2x)^2
  2 + 14x + 36x^2 + 40x^3 + 16x^4 = 2*(1+x)*(1+2x)^3

Cf. A185045, A208510.

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_] := u[n - 1, x] + 2 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[%]    (* A185045 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208659 *)
(* Using the function RiordanSquare defined in A321620 we also have: *)
A208659 = RiordanSquare[(1 + x)/(1 - x), 16] // Flatten (* Gerry Martens, Oct 16 2022 *)

u(n,x) = u(n-1,x) + 2x*v(n-1,x),
v(n,x) = u(n-1,x) + 2x*v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
As triangle T(n,k) with 0 <= k <= n: T(n,k) = A029653(n,k)*2^k. - Philippe Deléham, Mar 04 2012
Sum_{k=0..n} T(n,k)*x^k = 2*(1+x)*(1+2x)^(n-2) for n > 1. - Philippe Deléham, Mar 05 2012

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