A006138 -id:A006138 - cp's OEIS frontend

A210789 Triangle of coefficients of polynomials u(n,x) jointly generated with A210790; see the Formula section.

1, 1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 4, 8, 8, 5, 1, 5, 12, 18, 15, 8, 1, 6, 18, 32, 39, 28, 13, 1, 7, 24, 53, 77, 80, 51, 21, 1, 8, 32, 80, 142, 176, 160, 92, 34, 1, 9, 40, 116, 234, 352, 384, 312, 164, 55, 1, 10, 50, 160, 370, 632, 830, 812, 598, 290, 89, 1, 11, 60, 215

Offset: 1

Clark Kimberling, Mar 26 2012

Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).
Column 2: 1,2,3,4,5,6,7,8,...
Row sums: A006138.
Alternating row sums: signed Fibonacci numbers.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 28 2012

			First five rows:
  1;
  1, 1;
  1, 2, 2;
  1, 3, 4, 3;
  1, 4, 8, 8, 5;
First three polynomials u(n,x):
  1
  1 + x
  1 + 2x + 2x^2.
From _Philippe Deléham_, Mar 28 2012: (Start)
(1, 0, 0, -1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
  1;
  1, 0;
  1, 1, 0;
  1, 2, 2, 0;
  1, 3, 4, 3, 0;
  1, 4, 8, 8, 5, 0; (End)

Cf. A210790, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = 2; p = -1; f = 0;
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[%]    (* A210789 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210790 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A006138 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A105476 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x+2)*u(n-1,x) + (x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 28 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1+x-y*x-y*x^2-y^2*x^2)/(1-y*x-y*x^2-x^2-y^2*x^2).
T(n,k) = T(n-1,k-1) + T(n-2,k) + 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)

A210790 Triangle of coefficients of polynomials v(n,x) jointly generated with A210789; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 5, 5, 1, 4, 10, 10, 8, 1, 6, 14, 24, 20, 13, 1, 6, 21, 38, 52, 38, 21, 1, 8, 27, 65, 96, 109, 71, 34, 1, 8, 36, 92, 176, 224, 220, 130, 55, 1, 10, 44, 136, 280, 446, 500, 434, 235, 89, 1, 10, 55, 180, 440, 772, 1066, 1074, 839, 420, 144, 1

Offset: 1

Clark Kimberling, Mar 26 2012

Row n starts with 1 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Column 2: 2,2,4,4,6,6,8,8,...
Row sums: A105476.
Alternating row sums: signed Fibonacci numbers.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 28 2012

			First five rows:
  1;
  1,  2;
  1,  2,  3;
  1,  4,  5,  5;
  1,  4, 10, 10,  8;
First three polynomials v(n,x):
  1
  1 + 2x
  1 + 2x + 3x^2.
From _Philippe Deléham_, Mar 28 2012: (Start)
(1, 0, -1, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  2,  3,  0;
  1,  4,  5,  5,  0;
  1,  4, 10, 10,  8,  0; (End)

Cf. A210789, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = 2; p = -1; f = 0;
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[%]    (* A210789 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210790 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A006138 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A105476 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x+2)*u(n-1,x) + (x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 28 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1+x-y*x-y^2*x^2)/(1-y*x-x^2-y*x^2-y^2*x^2).
T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

A129710 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 2, 5, 1, 2, 7, 4, 2, 9, 9, 1, 2, 11, 16, 5, 2, 13, 25, 14, 1, 2, 15, 36, 30, 6, 2, 17, 49, 55, 20, 1, 2, 19, 64, 91, 50, 7, 2, 21, 81, 140, 105, 27, 1, 2, 23, 100, 204, 196, 77, 8, 2, 25, 121, 285, 336, 182, 35, 1, 2, 27, 144, 385, 540, 378, 112, 9, 2, 29, 169, 506

Offset: 0

Emeric Deutsch, May 12 2007

Also number of Fibonacci binary words of length n and having k 10 subwords.
Row n has 1+floor(n/2) terms.
Row sums are the Fibonacci numbers (A000045).
T(n,0)=2 for n >= 1.
Sum_{k>=0} k*T(n,k) = A023610(n-2).
Triangle, with zeros omitted, given by (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 14 2012
Riordan array ((1+x)/(1-x), x^2/(1-x)), zeros omitted. - Philippe Deléham, Jan 14 2012

			T(5,2)=4 because we have 10101, 01101, 01010 and 01011.
Triangle starts:
  1;
  2;
  2, 1;
  2, 3;
  2, 5, 1;
  2, 7, 4;
  2, 9, 9, 1;
Triangle (2, -1, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) begins:
  1;
  2, 0;
  2, 1, 0;
  2, 3, 0, 0;
  2, 5, 1, 0, 0;
  2, 7, 4, 0, 0, 0;
  2, 9, 9, 1, 0, 0, 0;

Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 200, flattened.)
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Thomas Grubb and Frederick Rajasekaran, Set Partition Patterns and the Dimension Index, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence.

Cf. A000045, A023610.
Cf. A029635, A029653.
Columns: A040000, A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.

T:=proc(n,k) if n=0 and k=0 then 1 elif k<=floor(n/2) then binomial(n-k,k)+binomial(n-k-1,k) else 0 fi end: for n from 0 to 18 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

MapAt[# - 1 &, #, 1] &@ Table[Binomial[n - k, k] + Binomial[n - k - 1, k], {n, 0, 16}, {k, 0, Floor[n/2]}] // Flatten (* Michael De Vlieger, Nov 15 2019 *)

T(n,k) = binomial(n-k,k) + binomial(n-k-1,k) for n >= 1 and 0 <= k <= floor(n/2).
G.f. = G(t,z) = (1+z)/(1-z-tz^2).
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A078050(n), A057079(n), A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively. - Philippe Deléham, Jan 14 2012
T(n,k) = T(n-1,k) + T(n-2,k-1) with T(0,0)=1, T(1,0)=2, T(1,1)=0 and T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Jan 14 2012

A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 5, 3, 0, 0, 8, 7, 1, 0, 0, 13, 15, 4, 0, 0, 0, 21, 30, 12, 1, 0, 0, 0, 34, 58, 31, 5, 0, 0, 0, 0, 55, 109, 73, 18, 1, 0, 0, 0, 0, 89, 201, 162, 54, 6, 0, 0, 0, 0, 0, 144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 10 2012

A skew version of A122075.

			Triangle begins :
  1
  2, 0
  3, 1, 0
  5, 3, 0, 0
  8, 7, 1, 0, 0
  13, 15, 4, 0, 0, 0
  21, 30, 12, 1, 0, 0, 0
  34, 58, 31, 5, 0, 0, 0, 0
  55, 109, 73, 18, 1, 0, 0, 0, 0
  89, 201, 162, 54, 6, 0, 0, 0, 0, 0
  144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A122075, A122950, A000045, A023610, A129707

T[0, 0] := 1; T[1, 0] := 2; T[1, 1] := 0; T[n_, k_] := T[n, k] = If[n<0, 0, If[k > n, 0, T[n - 1, k] + T[n - 2, k] + T[n - 2, k - 1]]]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 19 2017 *)

G.f.: (1+x)/(1-x-(1+y)*x^2).
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.

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A210789 Triangle of coefficients of polynomials u(n,x) jointly generated with A210790; see the Formula section.

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A210790 Triangle of coefficients of polynomials v(n,x) jointly generated with A210789; see the Formula section.

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A129710 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

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A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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