A000012 -id:A000012 - cp's OEIS frontend

A134249 Triangle read by rows, taken from the lower triangular matrix (M * A000012 + A000012 * M) - A000012; where M = lower triangular matrix with (1,1,1,...) in the main diagonal and the triangular numbers in the subdiagonal and A000012 = (1; 1,1; 1,1,1; ...).

1, 3, 1, 5, 7, 1, 8, 10, 13, 1, 12, 14, 17, 21, 1, 17, 19, 22, 26, 31, 1, 23, 25, 28, 32, 37, 43, 1, 30, 32, 35, 39, 44, 50, 57, 1, 38, 40, 43, 47, 52, 58, 65, 73, 1, 47, 49, 52, 56, 61, 67, 74, 82, 91, 1

Offset: 1

Gary W. Adamson, Oct 15 2007

Row sums = A037235: (1, 4, 13, 32, 65, 116, ...). Left column = A134250: (1, 3, 5, 8, 12, 17, 23, ...).

			First few rows of the triangle:
   1;
   3,  1;
   5,  7,  1;
   8, 10, 13,  1;
  12, 14, 17, 21,  1;
  17, 19, 22, 26, 31,  1;
  23, 25, 28, 32, 37, 43,  1;
  ...

Cf. A037235, A134250.

Edited by N. J. A. Sloane, Nov 01 2009

A134310 (A000012 * A134309 + A134309 * A000012) - A000012, where the sequences are interpreted as lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 8, 9, 11, 15, 16, 16, 17, 19, 23, 31, 32, 32, 33, 35, 39, 47, 63, 64, 64, 65, 67, 71, 79, 95, 127, 128, 128, 129, 131, 135, 143, 159, 191, 255, 256, 256, 257, 259, 263, 271, 287, 319, 383, 511

Offset: 0

Gary W. Adamson, Oct 19 2007

From M. F. Hasler, Mar 29 2022: (Start)
Both A000012 and A134309 have offset 0, so this triangular matrix also has row and column indices starting at 0.
Right (resp. left) multiplication by a diagonal matrix (such as A134309) amounts to multiplying the columns (resp. rows) of the other matrix by the diagonal elements. Therefore this matrix is the sum of the two lower triangular matrices with columns (resp. rows) filled with the same element given by sequence A134309 = (1, 1, 2, 4, 8, 16, ...), i. e., restricted to upper left 5 X 5 square:
           ( 1         )   ( 1         )   ( 1         )
           ( 1 1       )   ( 1 1       )   ( 1 1       )
  (this) = ( 1 1 2     ) + ( 2 2 2     ) - ( 1 1 1     ) .  (End)
           ( 1 1 2 4   )   ( 4 4 4 4   )   ( 1 1 1 1   )
           ( 1 1 2 4 8 )   ( 8 8 8 8 8 )   ( 1 1 1 1 1 )

			First few rows of the triangle:
   1;
   1,  1;
   2,  2,  3;
   4,  4,  5,  7;
   8,  8,  9, 11, 15;
  16, 16, 17, 19, 23, 31;
  32, 32, 33, 35, 39, 47, 63;
  ...

Cf. A000012 (all 1's), A134309 = diag(A011782 = 2^max(n-1,0), n >= 0), A000079.

Row sums are A134311.

A134310(r,c)=if(r>=c, 2^max(c-1,0)+2^max(r-1,0)-1)
matrix(8,8,i,j,A134310(i-1,j-1)) \\ M. F. Hasler, Mar 29 2022

(A000012 * A134309 + A134309 * A000012) - A000012, as infinite lower triangular matrices, where A000012 = (1; 1,1; 1,1,1; ...), and A134309 = diag(1, 1, 2, 4, 8, 16, ...) = diag(A011782 = 1 followed by 1, 2, 4, 8, ... = powers of 2).

Row sums: A134311 = (1, 2, 7, 20, 51, 122, 281, 632, ...).

Edited and offset corrected to 0 by M. F. Hasler, Mar 29 2022

A134464 (A127648 * A000012 + A000012 * A127773) - A000012.

Original entry on oeis.org

1, 2, 4, 3, 5, 8, 4, 6, 9, 13, 5, 7, 10, 14, 19, 6, 8, 11, 15, 20, 26, 7, 9, 12, 16, 21, 27, 34, 8, 10, 13, 17, 22, 28, 35, 43, 9, 11, 14, 18, 23, 29, 36, 44, 53, 10, 12, 15, 19, 24, 30, 37, 45, 54, 64

Offset: 1

Gary W. Adamson, Oct 26 2007

Row sums = A134465: (1, 6, 16, 32, 55, 86, ...).

			First few rows of the triangle:
  1;
  2,  4;
  3,  5,  8;
  4,  6,  9, 13;
  5,  7, 10, 14, 19;
  6,  8, 11, 15, 20, 26;
  7,  9, 12, 16, 21, 27, 34;
  ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A127648, A127773, A134465.

Flatten[Table[RecurrenceTable[{a[1]==i,a[n]==a[n-1]+n},a,{n,i}],{i,10}]] (* Harvey P. Dale, Nov 12 2013 *)

(A127648 * A000012 * A000012 * A127773) - A000012, as infinite lower triangular matrices.

A134674 A134673 * A000012.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 3, 4, 4, 5, 5, 5, 5, 5, 5, 3, 4, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 6, 5, 5, 8, 8, 8, 8, 8, 7, 7, 9, 9, 9, 9, 9, 9, 7, 5, 6, 6, 6, 10, 10, 10, 10, 10

Offset: 1

Gary W. Adamson, Nov 05 2007

Left column = A073757: (1, 2, 3, 4, 5, 5, 7, 8, 8, 7, ...).
Row sums = A134675: (1, 4, 9, 15, 25, 30, 49, 55, 80, ...).
n-th row (n>1) has n terms of "n", iff n is prime.

			First few rows of the triangle:
  1;
  2,  2;
  3,  3,  3;
  4,  3,  4,  4;
  5,  5,  5,  5,  5;
  5,  3,  4,  6,  6,  6;
  7,  7,  7,  7,  7,  7,  7;
  7,  6,  5,  5,  8,  8,  8,  8;
  8,  7,  7,  9,  9,  9,  9,  9,  9;
  7,  5,  6,  6,  6, 10, 10, 10, 10, 10;
  ...

Cf. A134673, A134675, A073757.

A134673 * A000012 as infinite lower triangular matrices. Triangle, partial sums of A134673 starting from the right of each row.

A134674(n,k) = Sum_{j=n-k+1..n} A134673(n,j).

A134699 Triangle read by rows: A051731^2 * A000012.

Original entry on oeis.org

1, 3, 1, 3, 1, 1, 6, 3, 1, 1, 3, 1, 1, 1, 1, 9, 5, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 10, 6, 3, 3, 1, 1, 1, 1, 6, 3, 3, 1, 1, 1, 1, 1, 1, 9, 5, 3, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 18, 12, 8, 5, 3, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Nov 06 2007

Left column = A007425.
Row sums = A007429: (1, 4, 5, 11, 7, 20, ...).
A134700 = A000012 * A127170.

			First few rows of the triangle:
   1;
   3, 1;
   3, 1, 1;
   6, 3, 1, 1;
   3, 1, 1, 1, 1;
   9, 5, 3, 1, 1, 1;
   3, 1, 1, 1, 1, 1, 1;
  10, 6, 3, 3, 1, 1, 1, 1;
  ...

Cf. A007425, A007429, A051731, A127170, A134700.

A051731^2 * A000012 = A127170 * A000012, as infinite lower triangular matrices.

More terms from Jinyuan Wang, Apr 29 2025

A135221 Triangle A007318 + A000012(signed) - I, I = Identity matrix, read by rows.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 4, 2, 1, 2, 3, 7, 3, 1, 0, 6, 9, 11, 4, 1, 2, 5, 16, 19, 16, 5, 1, 0, 8, 20, 36, 34, 22, 6, 1, 2, 7, 29, 55, 71, 55, 29, 7, 1, 0, 10, 35, 85, 125, 127, 83, 37, 8, 1, 2, 9, 46, 119, 211, 251, 211, 119, 46, 9, 1, 0, 12, 54, 166, 329, 463, 461, 331, 164, 56, 10, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

row sums = A051049: (1, 1, 4, 7, 16, 31, 64, ...).

			First few rows of the triangle:
  1;
  0, 1;
  2, 1,  1;
  0, 4,  2,  1;
  2, 3,  7,  3,  1;
  0, 6,  9, 11,  4,  1;
  2, 5, 16, 19, 16,  5,  1;
  0, 8, 20, 36, 34, 22,  6, 1;
  2, 7, 29, 55, 71, 55, 29, 7, 1;
...

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

Cf. A007318, A051049.

T:= function(n,k)
    if k=n then return 1;
    else return Binomial(n,k) + (-1)^(n-k);
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019

T:= func< n,k | k eq n select 1 else Binomial(n,k) +(-1)^(n-k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

seq(seq( `if`(k=n, 1, binomial(n,k) + (-1)^(n-k)), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

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

T(n,k) = if(k==n, 1, binomial(n,k) + (-1)^(n-k)); \\ G. C. Greubel, Nov 20 2019

def T(n, k):
    if (k==n): return 1
    else: return binomial(n,k) + (-1)^(n-k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

T(n,k) = A007318 + A000012(signed) - Identity matrix, where A000012(signed) = (1; -1,1; 1,-1,1; ...).

T(n,k) = (-1)^(n-k) + binomial(n,k), with T(n,n)=1. - G. C. Greubel, Nov 20 2019

More terms added by G. C. Greubel, Nov 20 2019

A135222 Triangle A049310 + A000012 - I, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 4, 1, 1, 1, 4, 1, 5, 1, 1, 2, 1, 7, 1, 6, 1, 1, 1, 5, 1, 11, 1, 7, 1, 1, 2, 1, 11, 1, 16, 1, 8, 1, 1, 1, 6, 1, 21, 1, 22, 1, 9, 1, 1, 2, 1, 16, 1, 36, 1, 29, 1, 10, 1, 1, 1, 7, 1, 36, 1, 57, 1, 37, 1, 11, 1, 1, 2, 1, 22, 1, 71, 1, 85, 1, 46, 1, 12, 1, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = A081659: (1, 2 4, 6, 9, 13, 19, 28, ...).

			First few rows of the triangle:
  1;
  1, 1;
  2, 1,  1;
  1, 3,  1,  1;
  2, 1,  4,  1,  1;
  1, 4,  1,  5,  1, 1;
  2, 1,  7,  1,  6, 1, 1;
  1, 5,  1, 11,  1, 7, 1, 1;
  2, 1, 11,  1, 16, 1, 8, 1, 1;
...

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

Cf. A049310, A081659.

T:= proc(n, k) option remember;
      if k=n then 1
    else 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) )
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..15); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==n, 1, 1 + Abs[Simplify[((1+(-1)^(n-k))/2)* Binomial[(n+k)/2, (n-k)/2]*Cos[(n-k)*Pi/2]]] ]; Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

@CachedFunction
def T(n, k):
    if (k==n): return 1
    else: return 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*pi/2) )
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Nov 20 2019

T(n,k) = A049310(n,k) + A000012(n,k) - Identity matrix, as infinite lower triangular matrices.

T(n,k) = 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) ), with T(n,n) = 1. - G. C. Greubel, Nov 20 2019

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

A135223 Triangle A000012 * A127648 * A103451, read by rows.

Original entry on oeis.org

1, 3, 2, 6, 2, 3, 10, 2, 3, 4, 15, 2, 3, 4, 5, 21, 2, 3, 4, 5, 6, 28, 2, 3, 4, 5, 6, 7, 36, 2, 3, 4, 5, 6, 7, 8, 45, 2, 3, 4, 5, 6, 7, 8, 9, 55, 2, 3, 4, 5, 6, 7, 8, 9, 10, 66, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 78, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 91, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

Offset: 1

Gary W. Adamson, Nov 23 2007

Row sums = A028387.

			First few rows of the triangle are:
   1;
   3, 2;
   6, 2, 3;
  10, 2, 3, 4;
  15, 2, 3, 4, 5;
...

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A002260, A103451, A127648.

T:= function(n,k)
    if k=1 then return Binomial(n+1,2);
    else return k;
    fi; end;
Flat(List([1..15], n-> List([1..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019

[k eq 1 select Binomial(n+1,2) else k: k in [1..n], n in [1..15]]; // G. C. Greubel, Nov 20 2019

seq(seq( `if`(k=1, binomial(n+1,2), k), k=1..n), n=1..15); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==1, Binomial[n+1, 2], k]; Table[T[n, k], {n, 15}, {k,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==1, binomial(n+1,2), k); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n,k):
    if (k==1): return binomial(n+1, 2)
    else: return k
[[T(n,k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Nov 20 2019

T(n,k) = A000012(n,k) * A127648(n,k) * A103451(n,k) as infinite lower triangular matrices. Replace left border of 1's in A002260 with (1, 3, 6, 10, 15, ...).

T(n, k) = k with T(n,1) = binomial(n+1, 2). - G. C. Greubel, Nov 20 2019

More terms added by G. C. Greubel, Nov 20 2019

A135224 Triangle A103451 * A007318 * A000012, read by rows. T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 9, 7, 4, 1, 17, 15, 11, 5, 1, 33, 31, 26, 16, 6, 1, 65, 63, 57, 42, 22, 7, 1, 129, 127, 120, 99, 64, 29, 8, 1, 257, 255, 247, 219, 163, 93, 37, 9, 1, 513, 511, 502, 466, 382, 256, 130, 46, 10, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = A132750: (1, 4, 9, 21, 49, 113, ...).

Left border = A083318: (1, 3, 5, 9, 17, 33, ...).

			First few rows of the triangle:
   1;
   3,  1;
   5,  3,  1;
   9,  7,  4,  1;
  17, 15, 11,  5,  1;
  33, 31, 26, 16,  6,  1;
  65, 63, 57, 42, 22,  7,  1;
...

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

Cf. A083318, A103451, A132750.

function T(n,k)
  if k eq 0 and n eq 0 then return 1;
  elif k eq 0 then return 2^n +1;
  else return (&+[Binomial(n, k+j): j in [0..n]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 20 2019

T:= proc(n, k) option remember;
      if k=0 and n=0 then 1
    elif k=0 then 2^n +1
    else add(binomial(n, k+j), j=0..n)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k] = If[k==n==0, 1, If[k==0, 2^n +1, Sum[Binomial[n, k + j], {j, 0, n}]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

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

def T(n, k):
    if (k==0 and n==0): return 1
    elif (k==0): return 2^n + 1
    else: return sum(binomial(n, k+j) for j in (0..n))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 20 2019

T(n, k) = A103451(n,k) * A007318(n,k) * A000012(n,k) as infinite lower triangular matrices.
T(n, k) = Sum_{j=0..n} binomial(n, k+j), with T(0,0) = 1 and T(n,0) = 2^n + 1. - G. C. Greubel, Nov 20 2019
T(n, k) = binomial(n, k)*hypergeom([1, k-n], [k+1], -1) - binomial(n, k+n+1)* hypergeom([1, k+1], [k+n+2], -1) + 0^k - 0^n. - Peter Luschny, Nov 20 2019

A135227 Triangle A000012 * A135225, read by rows.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 6, 4, 1, 6, 5, 10, 10, 5, 1, 7, 6, 15, 20, 15, 6, 1, 8, 7, 21, 35, 35, 21, 7, 1, 9, 8, 28, 56, 70, 56, 28, 8, 1, 10, 9, 36, 84, 126, 126, 84, 36, 9, 1, 11, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = A006127: (1, 3, 6, 11, 20, 37, ...).

			First few rows of the triangle:
  1;
  2, 1;
  3, 2,  1;
  4, 3,  3,  1;
  5, 4,  6,  4,  1;
  6, 5, 10, 10,  5, 1;
  7, 6, 15, 20, 15, 6, 1;
...

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

Cf. A006127, A007318, A135225.

T:= function(n,k)
    if k=0 then return 1;
    else return Binomial(n,k);
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019

[k eq 0 select n+1 else Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

seq(seq( `if`(k=0, n+1, binomial(n,k)), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

Table[If[k==0, n+1, Binomial[n, k]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==0, n+1, binomial(n,k)); \\ G. C. Greubel, Nov 20 2019

def T(n, k):
    if (k==0): return 1
    else: return binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019

A000012 * A135225 as infinite lower triangular matrices. Left border of 1's in Pascal's Triangle (A007318) is replaced with a column of (1,2,3,...).

T(n,k) = binomial(n,k), with T(n,0) = n+1. - G. C. Greubel, Nov 20 2019

More terms added by G. C. Greubel, Nov 20 2019

cp's OEIS Frontend

A134249 Triangle read by rows, taken from the lower triangular matrix (M * A000012 + A000012 * M) - A000012; where M = lower triangular matrix with (1,1,1,...) in the main diagonal and the triangular numbers in the subdiagonal and A000012 = (1; 1,1; 1,1,1; ...).

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A134310 (A000012 * A134309 + A134309 * A000012) - A000012, where the sequences are interpreted as lower triangular matrices.

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A134464 (A127648 * A000012 + A000012 * A127773) - A000012.

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A134674 A134673 * A000012.

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A134699 Triangle read by rows: A051731^2 * A000012.

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A135221 Triangle A007318 + A000012(signed) - I, I = Identity matrix, read by rows.

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A135222 Triangle A049310 + A000012 - I, read by rows.

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Maple

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Sage

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A135223 Triangle A000012 * A127648 * A103451, read by rows.