A127773 -id:A127773 - cp's OEIS frontend

A127777 A127773 * A002260 as infinite lower triangular matrices.

1, 3, 6, 6, 12, 18, 10, 20, 30, 40, 15, 30, 45, 60, 75, 21, 42, 63, 84, 105, 126, 28, 56, 84, 112, 140, 168, 196, 36, 72, 108, 144, 180, 216, 252, 288, 45, 90, 135, 180, 225, 270, 315, 360, 405, 55, 110, 165, 220, 275, 330, 385, 440, 495, 550, 66, 132, 198

Offset: 1

Gary W. Adamson, Jan 28 2007

Triangular number transform of A002260.

Swapped order of the factors: A002260 * A127773 = A127778.

			First few rows of the triangle:
   1;
   3,  6;
   6, 12, 18;
  10, 20, 30, 40;
  15, 30, 45, 60, 75;
  ...

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

Cf. A000217, A127773, A000537 (row sums), A127778.

T(n,k):=piecewise(k<=n,k*binomial(n+1,n-1),nMircea Merca, Apr 11 2012

Table[k*Binomial[n+1,n-1],{n,20},{k,n}]//Flatten (* Harvey P. Dale, Oct 26 2016 *)

T(n,k) = k*binomial(n+1,n-1) = Sum_{i=1..k} i*binomial(k,i)*binomial(n+2-k,n-i), 1 <= k <= n. - Mircea Merca, Apr 11 2012

More terms from Harvey P. Dale, Oct 26 2016

A143037 Triangle read by rows, A000012 * A127773 * A000012. A000012 is an infinite lower triangular matrix with all 1's, A127773 = (1; 0,3; 0,0,6; 0,0,0,10; ...).

Original entry on oeis.org

1, 3, 4, 6, 9, 10, 10, 16, 19, 20, 15, 25, 31, 34, 35, 21, 36, 46, 52, 55, 56, 28, 49, 64, 74, 80, 83, 84, 36, 64, 85, 100, 110, 116, 119, 120, 45, 81, 109, 130, 145, 155, 161, 164, 165, 55, 100, 136, 164, 185, 200, 210, 216, 219, 220

Offset: 1

Gary W. Adamson & Roger L. Bagula, Jul 18 2008

Right border = tetrahedral numbers, left border = triangular numbers.
Alternatively this is the square array A(n,k)
   1,   3,   6,  10,  15,  21,  28,  36,  45,  55, ...
   4,   9,  16,  25,  36,  49,  64,  81, 100, 121, ...
  10,  19,  31,  46,  64,  85, 109, 136, 166, 199, ...
  20,  34,  52,  74, 100, 130, 164, 202, 244, 290, ...
  35,  55,  80, 110, 145, 185, 230, 280, 335, 395, ...
  56,  83, 116, 155, 200, 251, 308, 371, 440, 515, ...
  ...
read by antidiagonals where A(n,k) = n*(n^2 + 3*k*n + 3*k^2 - 1)/6 is the sum of n triangular numbers starting at A000217(k). - R. J. Mathar, May 06 2015

			First few rows of the triangle:
   1;
   3,  4;
   6,  9, 10;
  10, 16, 19,  20;
  15, 25, 31,  34,  35;
  21, 36, 46,  52,  55,  56;
  28, 49, 64,  74,  80,  83,  84;
  36, 64, 85, 100, 110, 116, 119, 120;
  ...

Cf. A001296 (row sums).

A143037 := proc(n,k)
    k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6 ;
end proc:
seq(seq(A143037(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Aug 31 2022

T[n_,k_] = k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6;Table[T[n,k],{n,10},{k,n}]//Flatten (* James C. McMahon, Aug 13 2025 *)

T(n,k) = k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6. - R. J. Mathar, Aug 31 2022

A132818 The matrix product A127773 * A001263 of infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 3, 6, 18, 6, 10, 60, 60, 10, 15, 150, 300, 150, 15, 21, 315, 1050, 1050, 315, 21, 28, 588, 2940, 4900, 2940, 588, 28, 36, 1008, 7056, 17640, 17640, 7056, 1008, 36, 45, 1620, 15120, 52920, 79380, 52920, 15120, 1620, 45, 55, 2475, 29700, 138600, 291060

Offset: 1

Gary W. Adamson, Sep 02 2007

			First few rows of the triangle are:
1;
3, 3;
6, 18, 6;
10, 60, 60, 10;
15, 150, 300, 150, 15;
21, 315, 1050, 1050, 315, 21;
...

Cf. A127773, A001263, A002457 (row sums), A006472. A002695, A008459.

A132818 := proc(n,k)
    (n+1-k)*binomial(n+1,k)*binomial(n,k-1)/2 ;
end proc: # R. J. Mathar, Jul 29 2015

T(n,k) = A000217(n) * A001263(n,k).
Let a(n) = A006472(n), the 'triangular' factorial numbers. Then a(n)/(a(k)*a(n-k)) produces the present triangle (with a different offset). - Peter Bala, Dec 07 2011
T(n,k) = 1/2*(n+1-k)*C(n+1,k)*C(n,k-1), for n,k >= 1. O.g.f.: x*y/((1-x-x*y)^2 - 4*x^2*y)^(3/2) = x*y + x^2*(3*y + 3*y^2) + x^3*(6*y + 18*y^2 + 6*y^3) + .... Cf. A008459 with o.g.f.: x*y/((1-x-x*y)^2 - 4*x^2*y)^(1/2). Sum {k = 1..n-1} T(n,k)*2^(n-k) = A002695(n). - Peter Bala, Apr 10 2012

Corrected by R. J. Mathar, Jul 29 2015

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.

A127779 Triangle read by rows: A004736 * A127773.

Original entry on oeis.org

1, 2, 3, 3, 6, 6, 4, 9, 12, 10, 5, 12, 18, 20, 15, 6, 15, 24, 30, 30, 21, 7, 18, 30, 40, 45, 42, 28, 8, 21, 36, 50, 60, 63, 56, 36, 9, 24, 42, 60, 75, 84, 84, 72, 45

Offset: 1

Gary W. Adamson, Jan 28 2007

Row sums = bin(n,4), (A000332): (1, 5, 15, 35, ...).
From Clark Kimberling, Sep 16 2008: (Start)
As a rectangular array: R = A000027*A000217; R(m,n) = n*binomial(m+1,2).
R is the accumulation array (cf. A144112) of A002260 (rectangular, with n-th row (n,n,n,n,...)). (End)
As a rectangular array read by ascending antidiagonals, T(n,k) is the total number of triangles obtained when a triangle is cut into n parts with segments going down from the apex to its base and into k parts with segments parallel to its base. See Quora link. - Michel Marcus, Apr 07 2023

			First few rows of the triangle:
  1;
  2,  3;
  3,  6,  6;
  4,  9, 12, 10;
  5, 12, 18, 20, 15;
  6, 15, 24, 30, 30, 21;
  7, 18, 30, 40, 45, 42, 28;
  ...
First few rows of the rectangular array:
  1  3  6 10 15 ...
  2  6 12 20 30 ...
  3  9 18 30 45 ...
  4 12 24 40 60 ...
  5 15 30 50 75 ...
  ...

Quora, How many triangles are in this picture?.

Cf. A004736, A127773, A000217, A000332.

Cf. A002260. - Clark Kimberling, Sep 16 2008

A004736 * A127773 as infinite lower triangular matrices.

A131422 (A000012 * A127773) + (A127773 * A000012) - A000012.

Original entry on oeis.org

1, 3, 5, 6, 8, 11, 10, 12, 15, 19, 15, 17, 20, 24, 29, 21, 23, 26, 30, 35, 41, 28, 30, 33, 37, 42, 48, 55, 36, 38, 41, 45, 50, 56, 63, 71, 45, 47, 50, 54, 59, 65, 72, 80, 89, 55, 57, 60, 64, 69, 75, 82, 90, 99, 109

Offset: 1

Gary W. Adamson, Jul 10 2007

Left border = the triangular series, A000217. Right border = A028387, (1, 5, 11, 19, 29, 41, 55, 71, ...). Row sums = A131423: (1, 8, 25, 56, 105, 176, 273, ...).

			First few rows of the triangle are:
   1;
   3,  5;
   6,  8, 11;
  10, 12, 15, 19;
  15, 17, 20, 24, 29;
  21, 23, 26, 30, 35, 41;
  28, 30, 33, 37, 42, 48, 55;
  ...

Cf. A127773, A000217, A028387, A131423.

T:=proc(n,k) options operator, arrow: (1/2)*n*(n+1)+(1/2)*k*(k+1)-1 end proc: for n to 10 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form - Emeric Deutsch, Sep 06 2008

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

T(n,k) = (n^2 + n + k^2 + k - 2)/2 (1 <= k <= n). - Emeric Deutsch, Sep 06 2008

A132819 A001263 * A127773.

Original entry on oeis.org

1, 1, 3, 1, 9, 6, 1, 18, 36, 10, 1, 30, 120, 100, 15, 1, 45, 300, 500, 225, 21, 1, 63, 630, 1750, 1575, 441, 28, 1, 84, 1176, 4900, 7350, 4116, 784, 36, 1, 108, 2016, 11760, 26460, 24696, 9408, 1296, 45, 1, 135, 3240, 25200, 79380, 111132, 70560, 19440

Offset: 1

Gary W. Adamson, Sep 02 2007

Row sums = A132820: (1, 4, 16, 65, 266, ...).

			First few rows of the triangle are:
  1;
  1,  3;
  1,  9,   6;
  1, 18,  36,  10;
  1, 30, 120, 100,  15;
  1, 45, 300, 500, 225, 21;
  ...

Cf. A001263, A127773, A132820.

A001263 * A127773 as infinite lower triangular matrices.

A143219 Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.

Original entry on oeis.org

1, 2, 6, 3, 9, 18, 4, 12, 24, 40, 5, 15, 30, 50, 75, 6, 18, 36, 60, 90, 126, 7, 21, 42, 70, 105, 147, 196, 8, 24, 48, 80, 120, 168, 224, 288, 9, 27, 54, 90, 135, 189, 252, 324, 405, 10, 30, 60, 100, 150, 210, 280, 360, 450, 550

Offset: 1

Gary W. Adamson, Jul 30 2008

			First few rows of the triangle =
  1;
  2,  6;
  3,  9, 18;
  4, 12, 24, 40;
  5, 15, 30, 50,  75;
  6, 18, 36, 60,  90, 126;
  7, 21, 42, 70, 105, 147, 196;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000012, A127648, A127773.

Cf. A002024, A002411 (right border), A002414, A002417 (row sums), A011379.

[n*Binomial(k+1, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 12 2022

Table[n*Binomial[k+1, 2], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 12 2022 *)

flatten([[n*binomial(k+1, 2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 12 2022

Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.
Sum_{k=1..n} T(n, k) = A002417(n).
T(n, n) = A002411(n).
From G. C. Greubel, Jul 12 2022: (Start)
T(n, k) = A002024(n,k) * A127773(n,k).
T(n, k) = n * binomial(k+1, 2).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/4)*(4*n - 3*floor((n+1)/2) + 3)*binomial(2 + floor((n+1)/2), 3).
T(2*n-1, n) = A002414(n), n >= 1.
T(2*n-2, n-1) = A011379(n-1), n >= 2. (End)

A127775 Triangle read by rows: row n consists of n-1 zeros followed by 2n-1.

Original entry on oeis.org

1, 0, 3, 0, 0, 5, 0, 0, 0, 7, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 28 2007

a(A000217(n)) = A005408(n-1), T(n,n) = 2*n - 1. - Reinhard Zumkeller, Feb 11 2007

			First few rows of the triangle are:
1;
0, 3;
0, 0, 5;
0, 0, 0, 7;
...

Cf. A000217, A127648, A127773, A127774, A005408, A014480, A001787.

T(n,k) = (2*n - 1) * 0^(n - k), 1<=k<=n. - Reinhard Zumkeller, Feb 11 2007

More terms from Reinhard Zumkeller, Feb 11 2007

A127774 Triangle read by rows: row n consists of n-1 zeros followed by A000292(n).

Original entry on oeis.org

1, 0, 4, 0, 0, 10, 0, 0, 0, 20, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 56, 0, 0, 0, 0, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, 120, 0, 0, 0, 0, 0, 0, 0, 0, 165, 0, 0, 0, 0, 0, 0, 0, 0, 0, 220, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 286, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 364

Offset: 1

Gary W. Adamson, Jan 28 2007

Essentially triangle T(n,k), read by rows, given by (0,0,0,0,0,0,0,...) DELTA (4,-3/2,5/6,-1/3,3/5,-1/10,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 14 2011

			First few rows of the triangle are:
  1;
  0,  4;
  0,  0, 10;
  0,  0,  0, 20;
  0,  0,  0,  0, 35;
  ...

Cf. A000292, A127648, A127773.

from math import isqrt
from sympy.ntheory.primetest import is_square
def A127774(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(a+1)*(a+2)//6 if is_square((n<<3)+1) else 0 # Chai Wah Wu, Jun 09 2025

G.f.: 1/((x*y-1)^4). - R. J. Mathar, Aug 12 2015

More terms from Michel Marcus, Jun 10 2025

cp's OEIS Frontend

A127777 A127773 * A002260 as infinite lower triangular matrices.

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A143037 Triangle read by rows, A000012 * A127773 * A000012. A000012 is an infinite lower triangular matrix with all 1's, A127773 = (1; 0,3; 0,0,6; 0,0,0,10; ...).

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A132818 The matrix product A127773 * A001263 of infinite lower triangular matrices.

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

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A127779 Triangle read by rows: A004736 * A127773.

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A131422 (A000012 * A127773) + (A127773 * A000012) - A000012.

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A132819 A001263 * A127773.

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A143219 Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.

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