A210219 Triangle of coefficients of polynomials u(n,x) jointly generated with A210220; see the Formula section.

1, 2, 1, 3, 4, 1, 4, 9, 7, 1, 5, 16, 22, 11, 1, 6, 25, 50, 46, 16, 1, 7, 36, 95, 130, 86, 22, 1, 8, 49, 161, 295, 296, 148, 29, 1, 9, 64, 252, 581, 791, 610, 239, 37, 1, 10, 81, 372, 1036, 1792, 1897, 1163, 367, 46, 1, 11, 100, 525, 1716, 3612, 4900, 4166, 2083, 541, 56, 1

Offset

1,2

Comments

First two terms in row n: n,(n-1)^2; last term: 1.

Period of alternating row sums: (1,1,0).

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

Apparently this is A071920 without the marginal zeros, read by downwards antidiagonals, or T(n,k) = A071922(n,k). - R. J. Mathar, May 17 2014

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Formula

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

T(n,k) = Sum_{j=0..floor(k/2)} binomial(k,2*j)*binomial(n-j,k). - Detlef Meya, Dec 05 2023

Example

First five rows:
  1
  2...1
  3...4....1
  4...9....7....1
  5...16...22...11...1
First three polynomials u(n,x): 1, 2 + x, 3 + 4x + x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := 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[%]     (* A210219 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A210220 *)
(* alternative program *)
T[n_, k_] := Sum[Binomial[k, 2*j]*Binomial[n-j, k], {j, 0, Floor[k/2]}]; Flatten[Table[T[n, k], {n, 1, 11}, {k, 1, n}]] (* Detlef Meya, Dec 05 2023 *)

Prog

(PARI) T(n, k) = sum(j=0, k\2, binomial(k, 2*j)*binomial(n-j, k)) \\ Andrew Howroyd, Jan 01 2024

Crossrefs

Cf. A210220, A208510, A001906 (row sums).

Sequence in context: A103283 A104698 A067066 * A125103 A171275 A335312

Adjacent sequences: A210216 A210217 A210218 * A210220 A210221 A210222

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 19 2012

Status

approved