A133080 -id:A133080 - cp's OEIS frontend

A133084 A007318 * A133080.

1, 2, 1, 3, 2, 1, 4, 3, 4, 1, 5, 4, 10, 4, 1, 6, 5, 20, 10, 6, 1, 7, 6, 35, 20, 21, 6, 1, 8, 7, 56, 35, 56, 21, 8, 1, 9, 8, 84, 56, 126, 56, 36, 8, 1, 10, 9, 120, 84, 252, 126, 120, 36, 10, 1, 11, 10, 165, 120, 462, 252, 330, 120, 55, 10, 1

Offset: 1

Gary W. Adamson, Sep 16 2007

Row sums = A003945: (1, 3, 6, 12, 24, 48, 96, ...).

A133084 is jointly generated with A133567 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+v(n-1,x)+1. See the Mathematica section. - Clark Kimberling, Feb 28 2012

			First few rows of the triangle:
  1;
  2,  1;
  3,  2,  1;
  4,  3,  4,  1;
  5,  4, 10,  4,  1;
  6,  5, 20, 10,  6,  1;
  7,  6, 35, 20, 21,  6,  1;
  ...

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

Cf. A133080, A003945, A133567.

Cf. A000292 (column 3 and 4), A000389 (column 5 and 6), A000580 (column 7).

/* As triangle */ [[(1-(1+(-1)^k)/2 )*Binomial(n, k)+((1+(-1)^k)/2)*Binomial(n-1, k-1): k in [1..n]]: n in [1.. 11]]; // Vincenzo Librandi, Oct 21 2017

A133084 := proc(n,k)
    add(binomial(n-1,i-1)*A133080(i,k),i=1..n) ;
end proc: # R. J. Mathar, Jun 13 2025

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, 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[%]   (* A133567 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A133084 *)
(* Clark Kimberling, Feb 28 2012 *)
T[n_, k_] := If[k == n, 1, (1  - (1 + (-1)^k)/2 )*Binomial[n, k] + ((1 + (-1)^k)/2)*Binomial[n - 1, k - 1]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)

for(n=1,10, for(k=1,n, print1(if(k == n, 1, (1  - (1 + (-1)^k)/2 )*binomial(n, k) + ((1 + (-1)^k)/2)*binomial(n - 1, k - 1)), ", "))) \\ G. C. Greubel, Oct 21 2017

Binomial transform of triangle A133080.

A133081 An interpolation operator, companion to A133080.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Sep 09 2007

A133081 * [1,2,3,...] = A133090: (1, 1, 5, 3, 9, 5, 13, 7, 17, ...).

A133080: diagonal and subdiagonal are switched.

			First few rows of the triangle:
  1;
  1, 0;
  0, 1, 1;
  0, 0, 1, 0;
  0, 0, 0, 1, 1;
  ...

Cf. A133080, A133090, A040001 (row sums).

row(n) = vector(n, k, if (k==n-1, 1, if (k==n, n%2)));
lista(nn) = my(list=List()); for (n=1, nn, my(v=row(n)); for (k=1, #v, listput(list, v[k]))); Vec(list); \\ Michel Marcus, Mar 06 2022

Infinite lower triangular matrix, (1,0,1,0,...) in the main diagonal and (1,1,1,...) in the subdiagonal.

More terms from Michel Marcus, Mar 06 2022

A133599 A097806 * A133080 * A007318.

Original entry on oeis.org

1, 3, 1, 3, 3, 1, 3, 7, 5, 1, 3, 9, 10, 5, 1, 3, 13, 22, 18, 7, 1, 3, 15, 31, 34, 21, 7, 1, 3, 19, 51, 75, 65, 33, 9, 1, 3, 21, 64, 111, 120, 83, 36, 9, 1, 3, 25, 92, 196, 266, 238, 140, 52, 11, 1

Offset: 1

Gary W. Adamson, Sep 18 2007

Row sums = A133600: (1, 4, 7, 16, 28, 64, 112, ...).

			First few rows of the triangle:
  1;
  3,  1;
  3,  3,  1;
  3,  7,  5,  1;
  3,  9, 10,  5,  1;
  3, 13, 22, 18,  7,  1;
  ...

Cf. A097806, A133080, A133600.

A133599aux := proc(n,k)
    add(A097806(n,j)*A133080(j,k),j=k..n) ;
end proc:
A133599 := proc(n,k)
    add(A133599aux(n,j)*binomial(j-1,k-1),j=k..n) ;
end proc: # R. J. Mathar, Jun 20 2015

A097806 * A133080 * A007318 as infinite lower triangular matrices, where A097806 = the pairwise operator and A133080 = an interpolation operator.

A133083 A000012 * A133080.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1

Offset: 1

Gary W. Adamson, Sep 08 2007

Row sums = A032766, congruent to {0,1} (mod 3): (1, 3, 4, 6, 7, 9, 10, ...).

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  2, 1, 2, 1;
  2, 1, 2, 1, 1;
  2, 1, 2, 1, 2, 1;
  ...

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

Cf. A133080, A000012, A032766.

T[n_, k_] := If[k == n, 1, 1  + (1 - (-1)^k)/2 ]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)

for(n=1,10, for(k=1,n, print1(if(k==n, 1, 1 + (1-(-1)^k)/2), ", "))) \\ G. C. Greubel, Oct 21 2017

A000012 * A133080 as infinite lower triangular matrices.

A133087 A133080 * A007318.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 5, 4, 1, 1, 4, 6, 4, 1, 2, 9, 16, 14, 6, 1, 1, 6, 15, 20, 15, 6, 1, 2, 13, 36, 55, 50, 27, 8, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 2, 17, 64, 140, 196, 182, 112, 44, 10, 1

Offset: 0

Gary W. Adamson, Sep 08 2007

Row sums = A084221: (1, 3, 4, 12, 16, 48, 64, 192, ...).

Subtriangle of (0, 2, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

			First few rows of the triangle:
  1;
  2,  1;
  1,  2,  1;
  2,  5,  4,  1;
  1,  4,  6,  4,  1;
  2,  9, 16, 14,  6,  1;
  1,  6, 15, 20, 15,  6,  1;
  2, 13, 36, 55, 50, 27,  8,  1;
  1,  8, 28, 56, 70, 56, 28,  8,  1;
  ...
Triangle (0, 2, -3/2, -1/2, 0, 0, 0, ...) DELTA (1, 0, -1, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0,  2,  1;
  0,  1,  2,  1;
  0,  2,  5,  4,  1;
  0,  1,  4,  6,  4,  1;
  0,  2,  9, 16, 14,  6,  1;
  0,  1,  6, 15, 20, 15,  6,  1;
  0,  2, 13, 36, 55, 50, 27,  8,  1;
  0,  1,  8, 28, 56, 70, 56, 28,  8,  1;
  ...

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

Cf. A133080, A084221.

Columns: A000007, A114752, A133092.

CoefficientList[CoefficientList[Series[(1 + 2*x + y*x)/(1 - (1 + y)^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 21 2017 *)

A133080 * A007318 as infinite lower triangular matrices.
G.f.: (1+2*x+y*x)/(1-(1+y)^2*x^2). - Philippe Deléham, Mar 03 2012
T(n,k) = T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 2, T(1,1) = 1. - Philippe Deléham, Mar 03 2012
Sum_{k=0..n} T(n,k)*x^k = A059841(n), A019590(n+1), A000034(n), A084221(n), A133125(n) for x = -2, -1, 0, 1, 2 respectively. - Philippe Deléham, Mar 03 2012

A133091 A133080 * A002260.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 2, 4, 6, 4, 1, 2, 3, 4, 5, 2, 4, 6, 8, 10, 6, 1, 2, 3, 4, 5, 6, 7, 2, 4, 6, 8, 10, 12, 14, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2

Offset: 1

Gary W. Adamson, Sep 09 2007

Row sums = A133092: (1, 4, 6, 16, 15, 36, 28, ...).

			First few rows of the triangle:
  1;
  2,  2;
  1,  2,  3;
  2,  4,  6,  4;
  1,  2,  3,  4,  5;
  2,  4,  6,  8, 10,  6;
  1,  2,  3,  4,  5,  6,  7;
  ...

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

Cf. A133080, A002260, A133092.

T[n_, n_] := n; T[n_, k_] := (2 - (1 - (-1)^n)/2)*k; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)

for(n=1,10, for(k=1,n, print1(if(k==n, n,(2 - (1 - (-1)^n)/2)*k), ", "))) \\ G. C. Greubel, Oct 21 2017

A133080 * A002260 as infinite lower triangular matrices.

Odd n rows = (1,2,3,...,n). Even n rows = (2,4,6,...,n).

Corrected and extended by Philippe Deléham, Mar 02 2012

A133093 A007318 * A097806 * A133080.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 10, 6, 5, 1, 15, 10, 15, 5, 1, 21, 15, 35, 15, 7, 1, 28, 21, 70, 35, 28, 7, 1, 36, 28, 126, 70, 84, 28, 9, 1, 45, 36, 210, 126, 210, 84, 45, 9, 1, 55, 45, 330, 210, 462, 210, 165, 45, 11, 1

Offset: 1

Gary W. Adamson, Sep 09 2007

Row sums = A033484: (1, 4, 10, 22, 46, 94, ...).

			First few rows of the triangle:
   1;
   3,  1;
   6,  3,  1;
  10,  6,  5,  1;
  15, 10, 15,  5,  1;
  21, 15, 35, 15,  7,  1;
  28, 21, 70, 35, 28,  7,  1;
  ...

Cf. A133080, A097806, A033484. Duplicate of A131110.

A007318 * A097806 * A133080 as infinite lower triangular matrices.

Binomial transform of an infinite lower triangular matrix with (1,1,1,...) in the main diagonal, (2,1,2,1,...) in the subdiagonal and (1,0,1,0,...) in the subsubdiagonal.

A133805 Triangle read by rows: A007318 * A133566 * A133080.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 7, 6, 4, 1, 11, 10, 11, 5, 1, 16, 15, 25, 15, 6, 1, 22, 21, 50, 35, 22, 7, 1, 29, 28, 91, 70, 63, 28, 8, 1, 37, 36, 154, 126, 154, 84, 37, 9, 1, 46, 45, 246, 210, 336, 210, 129, 45, 10, 1, 56, 55, 375, 330, 672, 462, 375, 165, 56, 11, 1, 67, 66, 550, 495, 1254, 924, 957, 495, 231, 66, 12, 1

Offset: 1

Gary W. Adamson, Sep 23 2007

Row sums = A133806: (1, 3, 8, 18, 38, 78, 318, ...).

Left column = A000124.

			First few rows of the triangle:
   1;
   2,  1;
   4,  3,  1;
   7,  6,  4,  1;
  11, 10, 11,  5,  1;
  16, 15, 25, 15,  6,  1;
  22, 21, 50, 35, 22,  7,  1;
  29, 28, 91, 70, 63, 28,  8,  1;
  ...

Cf. A000124, A133080, A133566, A133804, A133806.

Binomial transform of (A133566 * A133080) where (A133566 * A133080) = an infinite lower triangular matrix with (1,1,1,...) in the main and subdiagonals and (1,0,1,0,1,...) in the subsubdiagonal.

a(21) = 1 inserted and more terms from Georg Fischer, Jun 08 2023

A133094 A007318 * A133080 * A097806, as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 7, 7, 5, 1, 9, 14, 14, 5, 1, 11, 25, 30, 16, 7, 1, 13, 41, 55, 41, 27, 7, 1, 15, 63, 91, 91, 77, 29, 9, 1, 17, 92, 140, 182, 182, 92, 44, 9, 1, 19, 129, 204, 336, 378, 246, 156, 46, 11, 1

Offset: 1

Gary W. Adamson, Sep 09 2007

Row sums give A133095.

Binomial transform of an infinite lower triangular matrix with (1,1,1,...) in the main diagonal, (2,1,2,1,...) in the subdiagonal and (0,1,0,1,...) in the subsubdiagonal.

			First few rows of the triangle:
   1;
   3,  1;
   5,  3,  1;
   7,  7,  5,  1;
   9, 14, 14,  5,  1;
  11, 25, 30, 16,  7,  1;
  13, 41, 55, 41, 27,  7,  1;
  ...

Cf. A133080, A097806, A133095.

A133113 A128174 * A007318 * A133080.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 4, 4, 1, 9, 6, 11, 4, 1, 12, 9, 24, 11, 6, 1, 16, 12, 46, 24, 22, 6, 1, 20, 16, 80, 46, 62, 22, 8, 1, 25, 20, 130, 80, 148, 62, 37, 8, 1, 30, 25, 200, 130, 314, 148, 128, 37, 10, 1

Offset: 1

Gary W. Adamson, Sep 14 2007

Row sums = 2^n - 1, A000225: (1, 3, 7, 15, 31, ...).

			First few rows of the triangle:
   1;
   2,  1;
   4,  2,  1;
   6,  4,  4,  1;
   9,  6, 11,  4,  1;
  12,  9, 24, 11,  6,  1;
  16, 12, 46, 24, 22,  6,  1;
  20, 16, 80, 46, 62, 22,  8,  1;
  ...

Cf. A133080, A007318, A128174, A000225.

A128174 * A007318 * A133080 as infinite lower triangular matrices.

cp's OEIS Frontend

A133084 A007318 * A133080.

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A133081 An interpolation operator, companion to A133080.

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A133599 A097806 * A133080 * A007318.

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A133083 A000012 * A133080.

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A133087 A133080 * A007318.

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A133091 A133080 * A002260.

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A133093 A007318 * A097806 * A133080.

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