A000012 -id:A000012 - cp's OEIS frontend

A135228 Triangle A000012(signed) * A007318 * A103451, read by rows.

1, 1, 1, 3, 1, 1, 5, 2, 2, 1, 11, 2, 4, 3, 1, 21, 3, 6, 7, 4, 1, 43, 3, 9, 13, 11, 5, 1, 85, 4, 12, 22, 24, 16, 6, 1, 171, 4, 16, 34, 46, 40, 22, 7, 1, 341, 5, 20, 50, 80, 86, 62, 29, 8, 1, 683, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 1365, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = A000975: (1, 2, 5, 10, 21, 42, 85, 170, ...).

Left border = A001045: (1, 1, 3, 5, 11, 21, 43, ...).

			First few rows of the triangle are:
    1;
    1, 1;
    3, 1,  1;
    5, 2,  2,  1;
   11, 2,  4,  3,  1;
   21, 3,  6,  7,  4,  1;
   43, 3,  9, 13, 11,  5,  1;
   85, 4, 12, 22, 24, 16,  6, 1;
  171, 4, 16, 34, 46, 40, 22, 7, 1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000975, A001045, A007318, A103451.

function T(n,k)
  if k eq 0 then return (2^(n+1) +(-1)^n)/3;
  else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=0 then (2^(n+1) +(-1)^n)/3
    else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==0, (2^(n+1) +(-1)^n)/3, Sum[Binomial[n-1-2*j, k-1], {j,0,Floor[(n-1)/2]}]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==0, (2^(n+1) +(-1)^n)/3, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) ); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n, k):
    if (k==0): return (2^(n+1) +(-1)^n)/3
    else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

T(n,k) = A000012(signed) * A007318 * A103451 as infinite lower triangular matrices. A000012(signed) = (1; -1,1; 1,-1,1; ...).
T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) =
(2^(n+1) - (-1)^(n+1))/3 (Jacobsthal_{n+1}).- G. C. Greubel, Nov 20 2019

Offset changed by G. C. Greubel, Nov 20 2019

A135229 Triangle A000012(signed) * A103451 * A007318, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 3, 6, 7, 4, 1, 1, 3, 9, 13, 11, 5, 1, 1, 4, 12, 22, 24, 16, 6, 1, 1, 4, 16, 34, 46, 40, 22, 7, 1, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

row sums = A005578 starting (1, 2, 3, 6, 11, 22, 43, 86, ...).

			First few rows of the triangle are:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 2,  4,  3,  1;
  1, 3,  6,  7,  4,  1;
  1, 3,  9, 13, 11,  5,  1;
  1, 4, 12, 22, 24, 16,  6, 1;
  1, 4, 16, 34, 46, 40, 22, 7, 1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A005578, A007318, A103451.

function T(n,k)
  if k eq 0 then return 1;
  else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=0 then 1
    else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==0, 1, Sum[Binomial[n-1-2*j, k-1], {j, 0, Floor[(n-1)/2]}]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==0, 1, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) ); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n, k):
    if (k==0): 1
    else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

T(n,k) = A000012(signed) * A103451 * A007318 as infinite lower triangular matrices, where A000012(signed) = (1; -1,1; 1,-1,1; ...).

T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = 1. - G. C. Greubel, Nov 20 2019

Offset changed by G. C. Greubel, Nov 20 2019

A135230 Triangle A103451 * A000012(signed) * A007318, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 4, 3, 1, 1, 3, 6, 7, 4, 1, 2, 3, 9, 13, 11, 5, 1, 1, 4, 12, 22, 24, 16, 6, 1, 2, 4, 16, 34, 46, 40, 22, 7, 1, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1, 2, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 1, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

row sums = A135231

			First few rows of the triangle are:
  1;
  1, 1;
  2, 1,  1;
  1, 2,  2,  1;
  2, 2,  4,  3,  1;
  1, 3,  6,  7,  4,  1;
  2, 3,  9, 13, 11,  5,  1;
  1, 4, 12, 22, 24, 16,  6, 1;
  2, 4, 16, 34, 46, 40, 22, 7, 1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000012, A007318, A103451, A135231.

function T(n,k)
  if k eq n then return 1;
  elif k eq 0 then return (3+(-1)^n)/2;
  else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=n then 1
    elif k=0 then (3+(-1)^n)/2
    else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3+(-1)^n)/2, Sum[Binomial[n-1 - 2*j, k-1], {j, 0, Floor[(n-1)/2]}] ]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==n, 1, if(k==0, (3+(-1)^n)/2, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) )); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n, k):
    if (k==n): return 1
    elif (k==0): return (3+(-1)^n)/2
    else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

T(n,k) = A103451 * A000012(signed) * A007318, where A000012(signed) = (1; -1,1; 1,-1,1;...).

T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = (3+(-1)^n)/2 and T(n,n) = 1. - G. C. Greubel, Nov 20 2019

More terms and offset changed by G. C. Greubel, Nov 20 2019

A135723 A122890 + A000012 - I, I = Identity matrix.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 11, 14, 1, 1, 1, 9, 71, 42, 1, 1, 1, 5, 161, 425, 132, 1, 1, 1, 2, 251, 1979, 2383, 429, 1, 1, 1, 1, 303, 6277, 19509, 12805, 1430, 1, 1, 1, 1, 299, 15675, 106493, 168609, 66947, 4862

Offset: 0

Gary W. Adamson, Nov 25 2007

Row sums = A005095: (1, 2, 4, 9, 28, 125, 726, ...).

Main diagonal = A000108, the Catalan numbers: (1, 1, 2, 5, 14, 42, 132, ...).

			First few rows of the triangle:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2,  5;
  1, 1, 1, 11,  14;
  1, 1, 1,  9,  71,   42;
  1, 1, 1,  5, 161,  425,  132;
  1, 1, 1,  2, 251, 1979, 2383, 429;
  ...

Cf. A122890, A005095, A000108.

A122890 + A000012 - Identity matrix; as infinite lower triangular matrices.

A135840 A135839 * A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 3, 2, 2, 1, 1, 4, 3, 2, 2, 1, 1, 4, 3, 3, 2, 2, 1, 1, 5, 4, 3, 3, 2, 2, 1, 1, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 8, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 1

Gary W. Adamson, Dec 01 2007

nonn

Row sums = A004652 starting (1, 3, 4, 7, 9, 13, 16, 21, ...).

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 2, 1, 1;
  3, 2, 2, 1, 1;
  4, 3, 2, 2, 1, 1;
  4, 3, 3, 2, 2, 1, 1;
  5, 4, 3, 3, 2, 2, 1, 1;
  ...

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

Cf. A135839, A004652, A135841.

T[1, 1] := 1; T[n_, 1] := Floor[(n + 2)/2]; T[n_, n_] := 1; T[n_, k_] := Floor[(n - k + 2)/2]; Table[T[n, k], {n, 1, 8}, {k, 1, n}]//Flatten (* G. C. Greubel, Dec 05 2016 *)

T(1, 1) = 1, T(n, 1) = floor((n + 2)/2), T(n, n) = 1, T(n, k) = floor((n - k + 2)/2). - G. C. Greubel, Dec 05 2016

A136536 Triangle read by rows: A001263 * A128064 * A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 5, 7, 3, 14, 19, 19, 4, 42, 51, 71, 41, 5, 132, 146, 216, 216, 76, 6, 429, 449, 617, 827, 547, 127, 7, 1430, 1457, 1793, 2675, 2675, 1205, 197, 8, 4862, 4897, 5497, 8017, 10369, 7429, 2389, 289, 9, 16796, 16840, 17830, 23770, 34858, 34858, 18226, 4366, 406, 10

Offset: 1

Gary W. Adamson, Jan 04 2008

Row sums = A001791: (1, 4, 15, 56, 210, 792, ...).

Left column = A000108 starting (1, 2, 5, 14, 42, 132, 429, ...).

			First few rows of the triangle:
    1;
    2,   2;
    5,   7,   3;
   14,  19,  19,   4;
   42,  51,  71,  41,   5;
  132, 146, 216, 216,  76,   6;
  429, 449, 617, 827, 547, 127,   7;
  ...

Cf. A000108, A001263, A001791, A128064.

a(46) = 16796 corrected and two more terms from Georg Fischer, May 31 2023

A136787 Triangle read by rows: A107131 * A000012.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 4, 1, 9, 9, 9, 7, 1, 21, 21, 21, 21, 11, 1, 51, 51, 51, 51, 46, 16, 1, 127, 127, 127, 127, 127, 92, 22, 1, 323, 323, 323, 323, 323, 309, 169, 29, 1, 835, 835, 835, 835, 835, 835, 709, 289, 37, 1

Offset: 1

Gary W. Adamson, Jan 21 2008

Row sums = A005773 starting (1, 2, 5, 13, 35, 96, 267, ...).
Leftmost column = A001006: (1, 1, 2, 4, 9, 21, 51, ...).
A136788 = A000012 * A107131.

			First few rows of the triangle:
    1;
    1,   1;
    2,   2,   1;
    4,   4,   4,   1;
    9,   9,   9,   7,   1;
   21,  21,  21,  21,  11,  1;
   51,  51,  51,  51,  46, 16,  1;
  127, 127, 127, 127, 127, 92, 22, 1;
  ...

Cf. A005773, A001006, A136788.

A107131 * A000012 as infinite lower triangular matrices.

A136788 Triangle read by rows: A000012 * A107131.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 7, 1, 1, 2, 6, 17, 11, 1, 1, 2, 6, 22, 41, 16, 1, 1, 2, 6, 22, 76, 86, 22, 1, 1, 2, 6, 22, 90, 226, 162, 29, 1, 1, 2, 6, 22, 90, 352, 582, 281, 37, 1

Offset: 0

Gary W. Adamson, Jan 21 2008

Row sums = A086615: (1, 2, 4, 8, 17, 38, ...).

A136787 = A107131 * A000012.

			First few rows of the triangle:
  1;
  1, 1;
  1, 2, 1;
  1, 2, 4,  1;
  1, 2, 6,  7,  1;
  1, 2, 6, 17, 11,  1;
  1, 2, 6, 22, 41, 16,  1;
  1, 2, 6, 22, 76, 86, 22, 1;
  ...

Cf. A086615, A136787.

A000012 * A107131 as infinite lower triangular matrices.

A137593 Triangle read by rows: matrix product A000012 * A136717.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 4, 16, 12, 1, 4, 16, 84, 48, 1, 4, 16, 156, 504, 192, 1, 4, 16, 156, 1464, 3312, 960, 1, 4, 16, 156, 1464, 14112, 24720, 5760, 1, 4, 16, 156, 1464, 24912, 158640, 189360, 34560, 1, 4, 16, 156, 1464, 24912, 400560, 1761840, 1607040, 241920

Offset: 1

Gary W. Adamson, Jan 28 2008

Right border = A004527: (1, 2, 4, 12, 48, 192, ...).

Row sums = A007489 starting (1, 3, 9, 33, 153, ...).

			First few rows of the triangle:
  1;
  1, 2;
  1, 4,  4;
  1, 4, 16,  12;
  1, 4, 16,  84,   48;
  1, 4, 16, 156,  504,  192;
  1, 4, 16, 156, 1464, 3312, 960;
  ...

Cf. A136717, A004527, A007489.

A000012 * A136717 as infinite lower triangular matrices.

Typo in eighth row corrected by Olivier Gérard, Oct 30 2012

A137633 Triangle read by rows, A000012 * A026794.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 7, 2, 1, 1, 12, 3, 1, 1, 1, 19, 5, 2, 1, 1, 1, 30, 7, 3, 1, 1, 1, 1, 45, 11, 4, 2, 1, 1, 1, 1, 67, 15, 6, 3, 1, 1, 1, 1, 1, 97, 22, 8, 4, 2, 1, 1, 1, 1, 1, 139, 30, 11, 5, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Jan 31 2008

Left border = A000070: (1, 2, 4, 7, 12, 19, 30, 45, ...).
Second column = A000041, the partition numbers: (1, 1, 2, 3, 5, 7, 11, ...).
Row sums = A016905: (1, 3, 6, 11, 18, 29, 44, ...).

			First few rows of the triangle:
   1;
   2, 1;
   4, 1, 1;
   7, 2, 1, 1;
  12, 3, 1, 1, 1;
  19, 5, 2, 1, 1, 1;
  30, 7, 3, 1, 1, 1, 1;
  ...

Cf. A000041, A000070, A016905, A026794.

A000012 * A026794 as infinite lower triangular matrices.

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A135228 Triangle A000012(signed) * A007318 * A103451, read by rows.

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A135229 Triangle A000012(signed) * A103451 * A007318, read by rows.

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A135230 Triangle A103451 * A000012(signed) * A007318, read by rows.

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A135723 A122890 + A000012 - I, I = Identity matrix.

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A135840 A135839 * A000012 as infinite lower triangular matrices.

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A136536 Triangle read by rows: A001263 * A128064 * A000012 as infinite lower triangular matrices.

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A136787 Triangle read by rows: A107131 * A000012.

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