A056242 -id:A056242 - cp's OEIS frontend

A165241 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

1, 1, 1, 2, 3, 1, 4, 9, 6, 1, 8, 24, 25, 10, 1, 16, 60, 85, 55, 15, 1, 32, 144, 258, 231, 105, 21, 1, 64, 336, 728, 833, 532, 182, 28, 1, 128, 768, 1952, 2720, 2241, 1092, 294, 36, 1, 256, 1728, 5040, 8280, 8361, 5301, 2058, 450, 45, 1

Offset: 0

Philippe Deléham, Sep 09 2009

Rows sums: A006012; Diagonal sums: A052960.

The sums of each column of A117317 with its subsequent column, treated as a lower triangular matrix with an initial null column attached, or, equivalently, the products of the row polynomials p(n,y) of A117317 with (1+y) with the initial first row below added to the final result. The reversal of A117317 is A056242 with several combinatorial interpretations. - Tom Copeland, Jan 08 2017

			Triangle begins:
  1;
  1,  1;
  2,  3,  1;
  4,  9,  6,  1;
  8, 24, 25, 10,  1; ...

Cf. A000012, A000217, A005582, A011782, A084858.

Cf. A056242, A117317.

Sum_{k=0..n} T(n,k)*x^k = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. Sum_{k=0..n} T(n,k)*x^(n-k) = A123335(n), A000007(n), A000012(n), A006012(n), A084120(n), A090965(n), A165225(n), A165229(n), A165230(n), A165231(n), A165232(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively.
G.f.: (1-(1+y)*x)/(1-2(1+y)*x+(y+y^2)*x^2). - Philippe Deléham, Dec 19 2011
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2) with T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if nPhilippe Deléham, Dec 19 2011

O.g.f. corrected by Tom Copeland, Jan 15 2017

A210754 Triangle of coefficients of polynomials v(n,x) jointly generated with A210753; see the Formula section.

Original entry on oeis.org

1, 3, 2, 6, 9, 4, 10, 25, 24, 8, 15, 55, 85, 60, 16, 21, 105, 231, 258, 144, 32, 28, 182, 532, 833, 728, 336, 64, 36, 294, 1092, 2241, 2720, 1952, 768, 128, 45, 450, 2058, 5301, 8361, 8280, 5040, 1728, 256, 55, 660, 3630, 11385, 22363, 28610, 23920

Offset: 1

Clark Kimberling, Mar 25 2012

Column 1: triangular numbers, A000217
Coefficient of v(n,x):  2^(n-1)
Row sums:  A035344
Alternating row sums: 1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
Appears to be the reversed row polynomials of A165241 with the unit diagonal removed. If so, the o.g.f. is [1-(1+y)x]/[1-2(1+y)x+(1+y)x^2] - 1/(1-x) and the triangular matrix here may be formed by adding each column of the matrix of A056242, presented in the example section with the additional zeros, to its subsequent column with the first row ignored. - Tom Copeland, Jan 09 2017

			First five rows:
1
3....2
6....9....4
10...25...24...8
15...55...85...60...16
First three polynomials v(n,x): 1, 3 + 2x, 6 + 9x +4x^2

Cf. A210753, A208510.

Cf. A056242, A165241.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*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[%]    (* A210753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210754 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007070 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A035344 *)

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

cp's OEIS Frontend

A165241 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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