A000012 -id:A000012 - cp's OEIS frontend

A144337 Triangle read by rows, A000012 * (2*A144328 - A000012).

1, 2, 1, 3, 2, 3, 4, 3, 6, 5, 5, 4, 9, 10, 7, 6, 5, 12, 15, 14, 9, 7, 6, 15, 20, 21, 18, 11, 8, 7, 18, 25, 28, 27, 22, 13, 9, 8, 21, 30, 35, 36, 33, 26, 15, 10, 9, 24, 35, 42, 45, 44, 39, 30, 17, 11, 10, 27, 40, 49, 54, 55, 52, 45, 34, 19

Offset: 1

Gary W. Adamson and Roger L. Bagula, Sep 18 2008

When the first column is removed from this triangle (A144337), the result is triangle A101447. - Georg Fischer, Aug 10 2023

			First few rows of the triangle:
   1;
   2,  1;
   3,  2,  3;
   4,  3,  6,  5;
   5,  4,  9, 10,  7;
   6,  5, 12, 15, 14,  9;
   7,  6, 15, 20, 21, 18, 11;
   8,  7, 18, 25, 28, 27, 22, 13;
   9,  8, 21, 30, 35, 36, 33, 26, 15;
  10,  9, 24, 35, 42, 45, 44, 39, 30, 17;
  ...

Cf. A081489, A101447, A144328.

a(25) corrected by Georg Fischer, Aug 10 2023

A144693 Triangle read by rows, A000012 * (3A144328 - 2A000012), where A000012 means a lower triangular matrix of all 1's.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 4, 3, 8, 7, 5, 4, 12, 14, 10, 6, 5, 16, 21, 20, 13, 7, 6, 20, 28, 30, 26, 16, 8, 7, 24, 35, 40, 39, 32, 19, 9, 8, 28, 42, 50, 52, 48, 38, 22, 10, 9, 32, 49, 60, 65, 64, 57, 44, 25, 11, 10, 36, 56, 70, 78, 80, 76, 66, 50, 28

Offset: 1

Gary W. Adamson, Sep 19 2008

			Partial sums by columns of the triangle (3*A144328 - 2*A000012):
  1;
  1, 1;
  1, 1, 4;
  1, 1, 4, 7;
  1, 1, 4, 7, 10;
  ...
First few rows of the triangle:
  1;
  2, 1
  3, 2,  4;
  4, 3,  8,  7;
  5, 4, 12, 14, 10;
  6, 5, 16, 21, 20, 13;
  7, 6, 20, 28, 30, 26, 16;
  8, 7, 24, 35, 40, 39, 32, 19;
  ...

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

Cf. A000012, A064808, A144328.

A144693:= func< n,k | k eq 1 select n else (3*k-5)*(n-k+1) >;
[A144693(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 19 2021

T[n_, k_]:= (3*k -5 +3*Boole[k==1])*(n-k+1);
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 19 2021 *)

def A144693(n,k): return (3*k -5 +3*bool(k==1))*(n-k+1)
flatten([[A144693(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 19 2021

Sum_{k=1..n} T(n, k) = A064808(n).

T(n, k) = (3*k -5 +3*[k=1])*(n-k+1). - G. C. Greubel, Oct 19 2021

A145972 Triangle read by rows, A000012 * A053121.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 3, 4, 1, 1, 4, 8, 4, 5, 1, 1, 9, 8, 13, 5, 6, 1, 1, 9, 22, 13, 19, 6, 7, 1, 1, 23, 22, 41, 19, 26, 7, 8, 1, 1, 23, 64, 41, 67, 26, 34, 8, 9, 1, 1, 65, 64, 131, 67, 101, 34, 43, 9, 10, 1, 1

Offset: 0

Gary W. Adamson & Roger L. Bagula, Oct 25 2008

Row sums = A036256: (1, 2, 4, 7, 13, 23, 43,...).

			A053121 starts:
1;
0, 1;
1, 0, 1;
0, 2, 0, 1;
...
Taking partial sums from the top, we get A145972:
1;
1, 1;
2, 1, 1;
2, 3, 1, 1;
4, 3, 4, 1, 1;
4, 8, 4, 5, 1, 1;
9, 8, 13, 5, 6, 1, 1;
9, 22, 13, 19, 6, 7, 1, 1;
23, 22, 41, 19, 26, 7, 8, 1, 1;
23, 64, 41, 67, 26, 34, 8, 9, 1, 1;
...

Cf. A036256, A053121

Triangle read by rows, A000012 * A053121, = partial sums of column terms of triangle A053121.

A152194 Triangle read by rows, A034839 * A000012.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 8, 7, 1, 16, 15, 5, 32, 31, 16, 1, 64, 63, 42, 7, 128, 127, 99, 29, 1256, 255, 219, 93, 9, 512, 511, 466, 256, 46, 1, 1024, 1023, 968, 638, 176, 11, 2048, 2047, 1981, 1486, 562, 67, 1

Offset: 0

Gary W. Adamson, Nov 28 2008

Row sums = A045891: (1, 1, 3, 7, 16, 36, 80, 176,...)

			First few rows of the triangle =
1;
1;
2, 1;
4, 3;
8, 7, 1;
16, 15, 5;
32, 31, 16, 1;
64, 63, 42, 7;
128, 127, 99, 29, 1;
256, 255, 219, 93, 9;
512, 511, 466, 256, 46, 1;
1024, 1023, 968, 638, 176, 11;
2048, 2047, 1981, 1486, 562, 67, 1;
...

A034839, A045891

A034839 * A000012 = partial sums of A034839 by rows, starting from the right.

A153345 Triangle read by rows, A000012 * A055830.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 7, 4, 1, 0, 12, 11, 5, 1, 0, 20, 26, 17, 6, 1, 0, 33, 56, 48, 24, 7, 1, 0, 54, 114, 121, 78, 32, 8, 1, 0, 88, 223, 283, 223, 117, 41, 9, 1, 0, 143, 424, 627, 584, 372, 166, 51, 10, 1, 0, 232, 789, 1334, 1434, 1073, 579, 226, 62, 11, 1, 0, 376, 1444, 2750, 3352, 2879, 1818, 856, 298, 74, 12, 1, 0

Offset: 1

Gary W. Adamson, Dec 24 2008

Row sums = Pell numbers, A000129: (1, 2, 5, 12, 29,...).

Left border = A000071.

			First few rows of the triangle =
1;
2, 0;
4, 1, 0;
7, 4, 1, 0;
12, 11, 5, 1, 0;
20, 26, 17, 6, 1, 0;
33, 56, 48, 24, 7, 1, 0;
54, 114, 121, 78, 32, 8, 1, 0;
88, 223, 283, 223, 117, 41, 9, 1, 0;
143, 424, 627, 584, 372, 166, 51, 10, 1, 0;
232, 789, 1334, 1434, 1073, 579, 226, 62, 11, 1, 0;
...

Cf. A000129, A055830.

Triangle read by rows, A000012 * triangle A055830 = partial sums of A055830 columns.

a(21) = 0 inserted and more terms from Georg Fischer, May 29 2023

A153346 Triangle read by rows: A000012 * A153345.

Original entry on oeis.org

1, 3, 0, 7, 1, 0, 14, 5, 1, 0, 26, 16, 6, 1, 0, 46, 42, 23, 7, 1, 0, 79, 98, 71, 31, 8, 1, 0, 133, 212, 192, 109, 40, 9, 1, 0, 221, 435, 475, 332, 157, 50, 10, 1, 0, 364, 859, 1102, 916, 529, 216, 61, 11, 1, 0, 596, 1648, 2436, 2350, 1602, 795, 287, 73, 12, 1, 0, 972, 3092, 5186, 5702, 4481, 2613, 1143, 371, 86, 13, 1, 0

Offset: 0

Gary W. Adamson, Dec 24 2008

Row sums = A048739: (1, 3, 8, 20, 49, 119, 288, ...).

Left border = A001924.

			First few rows of the triangle:
    1;
    3,    0;
    7,    1,    0;
   14,    5,    1,    0;
   26,   16,    6,    1,    0;
   46,   42,   23,    7,    1,   0
   79,   98,   71,   31,    8,   1,   0;
  133,  212,  192,  109,   40,   9,   1,  0;
  221,  435,  475,  332,  157,  50,  10,  1,  0;
  364,  859, 1102,  916,  529, 216,  61, 11,  1, 0;
  596, 1648, 2436, 2350, 1602, 795, 287, 73, 12, 1, 0;
  ...

Cf. A001924, A048739, A153345.

a(44) = 0 corrected and more terms from Georg Fischer, Jun 05 2023

A157901 Triangle read by rows: A000012 * A157898.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 3, 6, 10, 8, 8, 3, 9, 16, 24, 16, 16, 4, 12, 28, 40, 56, 32, 32, 4, 16, 40, 80, 96, 128, 64, 64, 5, 20, 60, 120, 216, 224, 288, 128, 128, 5, 25, 80, 200, 336, 560, 512, 640, 256, 256, 6, 30, 110, 280, 616, 896, 1408, 1152, 1408, 512, 512

Offset: 0

Gary W. Adamson & Roger L. Bagula, Mar 08 2009

Multiplication of the lower triangular matrix A157898 from the left by A000012 means: these are partial column sums of A157898.

			First few rows of the triangle, n>=0:
  1;
  1,  1;
  2,  2,  2;
  2,  4,  4,   4;
  3,  6, 10,   8,   8;
  3,  9, 16,  24,  16,  16;
  4, 12, 28,  40,  56,  32,  32;
  4, 16, 40,  80,  96, 128,  64,  64;
  5, 20, 60, 120, 216, 224, 288, 128, 128;
  5, 25, 80, 200, 336, 560, 512, 640, 256, 256;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000012, A105635, A157898.
Columns: A004526 (k=0), A002620 (k=1), A006584 (k=2), 4*A096338 (k=3), 8*A177747 (k=4), 16*A299337 (k=5), 32*A178440 (k=6).
Sums include: A105635(n+1) (row), A166486(n+1) (alternating sign diagonal), A232801(n+1) (diagonal).

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function t(n, k) // t = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
  end if; return t;
end function;
A157898:= func< n, k | (&+[(-1)^(n-j)*Binomial(n, j)*t(j, k): j in [k..n]]) >;
A157071:= func< n,k | (&+[A157898(j+k,k): j in [0..n-k]]) >;
[A157071(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 27 2025

t[n_, k_]:= t[n,k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1,k-1] +2*t[n-1,k] - (2 -Boole[n==2])*t[n-2,k-1]]; (* t = A059576 *)
A157898[n_, k_]:= A157898[n,k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];
A157901[n_, k_]:= A157901[n,k]= Sum[A157898[j+k,k], {j,0,n-k}];
Table[A157901[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 27 2025 *)

@CachedFunction
def t(n, k): # t = A059576
    if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
    else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
def A157898(n, k): return sum((-1)^(n+k-j)*binomial(n, j+k)*t(j+k, k) for j in range(n-k+1))
def A157071(n,k): return sum(A157898(j+k,k) for j in range(n-k+1))
print(flatten([[A157071(n,k) for k in range(n+1)] for n in range(10)])) # G. C. Greubel, Aug 27 2025

T(n,k) = Sum_{j=0..n} A157898(j,k).

Edited by the Associate Editors of the OEIS, Apr 10 2009

More terms from G. C. Greubel, Aug 27 2025

A158822 Triangle read by rows, matrix triple product A000012 * A145677 * A000012.

Original entry on oeis.org

1, 3, 1, 6, 3, 2, 10, 6, 5, 3, 15, 10, 9, 7, 4, 21, 15, 14, 12, 9, 5, 28, 21, 20, 18, 15, 11, 6, 36, 28, 27, 25, 22, 18, 13, 7, 45, 36, 35, 33, 30, 26, 21, 15, 8, 55, 45, 44, 42, 39, 35, 30, 24, 17, 9, 66, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10

Offset: 0

Gary W. Adamson and Roger L. Bagula, Mar 28 2009

			First few rows of the triangle =
   1;
   3,  1;
   6,  3,  2;
  10,  6,  5,  3;
  15, 10,  9,  7,  4;
  21, 15, 14, 12,  9,  5;
  28, 21, 10, 18, 15, 11,  6;
  36, 28, 27, 25, 22, 18, 13,  7;
  45, 36, 35, 33, 30, 26, 21, 15,  8;
  55, 45, 44, 42, 39, 35, 30, 24, 17,  9;
  66, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10;
  78, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11;
  91, 78, 77, 75, 72, 68, 63, 57, 50, 42, 33, 23, 12;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000012, A000096, A000217, A006527, A034828, A055998, A145677, A134479.

T[n_, k_]:= If[k==0, Binomial[n+2, 2], (n+1-k)*(n+k)/2];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 26 2021 *)

def A158822(n,k):
    if (k==0): return binomial(n+2, 2)
    else: return (n-k+1)*(n+k)/2
flatten([[A158822(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 26 2021

Triangle read by rows, A000012 * A145677 * A000012; where A000012 = an infinite lower triangular matrix: (1; 1,1; 1,1,1; ...), with all 1's.
From G. C. Greubel, Dec 26 2021: (Start)
T(n, k) = (n+1-k)*(n+k)/2 with T(n, 0) = binomial(n+2, 2).
Sum_{k=0..n} T(n, k) = (1/3)*(n+1)*(n^2 + 2*n + 3) = A006527(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = binomial(n+2, 2) + A034828(n+1).
T(n, 1) = A000217(n).
T(n, 2) = A000096(n-1).
T(n, 3) = A055998(n-2).
T(2*n, n) = A134479(n). (End)

Definition corrected by Michael Somos, Nov 05 2011

A158946 Triangle read by rows, A000012(signed) * A145677 * A000012.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 3, 3, 2, 3, 1, 4, 3, 3, 2, 4, 1, 5, 4, 3, 4, 2, 5, 1, 6, 4, 4, 3, 5, 2, 6, 1, 7, 5, 4, 5, 3, 6, 2, 7, 1, 8, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 6, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 6, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 11

Offset: 1

Gary W. Adamson, Mar 31 2009

Row sums = A000982 starting with offset 1: (1, 2, 5, 8, 13, 18, 25,...).

			First few rows of the triangle =
1;
1, 1;
2, 1, 2;
2, 2, 1, 3;
3, 2, 3, 1, 4;
3, 3, 2, 4, 1, 5;
4, 3, 4, 2, 5, 1, 6;
4, 4, 3, 5, 2, 6, 1, 7;
5, 4, 5, 3, 6, 2, 7, 1, 8;
5, 5, 4, 6, 3, 7, 2, 8, 1, 9;
6, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10;
6, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 11;
7, 6, 7, 5, 8, 4, 9, 3, 10, 2, 11, 1, 12;
7, 7, 6, 8, 5, 9, 4, 10, 3, 11, 2, 12, 1, 13;
...

Cf. A145677, A000982

Triangle read by rows, A000012(signed) * A145677 * A000012. A000012(signed) = an infinite lower triangular matrix with (1,-1,1,-1,...) in every column. A145677 = an infinite lower triangular matrix with all 1's as the left border, right border = (1, 1, 2, 3, 4, 5,...), and the rest zeros.

A166556 Triangle read by rows, A000012 * A047999.

Original entry on oeis.org

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

Offset: 0

Gary W. Adamson, Oct 17 2009

			First few rows of the triangle =
   1;
   2, 1;
   3, 1, 1;
   4, 2, 2, 1;
   5, 2, 2, 1, 1;
   6, 3, 2, 1, 2, 1;
   7, 3, 3, 1, 3, 1, 1;
   8, 4, 4, 2, 4, 2, 2, 1;
   9, 4, 4, 2, 4, 2, 2, 1, 1;
  10, 5, 4, 2, 4, 2, 2, 1, 2, 1;
  11, 5, 5, 2, 4, 2, 2, 1, 3, 1, 1;
  12, 6, 6, 3, 4, 2, 2, 1, 4, 2, 2, 1;
  13, 6, 6, 3, 5, 2, 2, 1, 5, 2, 2, 1, 1;
  ...

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

Cf. A000027, A004524, A004526, A047999, A080100.

Sums include: A006046 (row), A007729 (diagonal).

A166556:= func< n,k | (&+[(Binomial(j,k) mod 2): j in [k..n]]) >;
[A166556(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 02 2024

A166556 := proc(n,k)
    local j;
    add(A047999(j,k),j=k..n) ;
end proc: # R. J. Mathar, Jul 21 2016

A166556[n_, k_]:= Sum[Mod[Binomial[j,k], 2], {j,k,n}];
Table[A166556[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2024 *)

def A166556(n,k): return sum(binomial(j,k)%2 for j in range(k,n+1))
print(flatten([[A166556(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 02 2024

Triangle read by rows, A000012 * A047999; where A000012 = an infinite lower triangular matrix with all 1's: [1; 1,1; 1,1,1;..]; and A047999 = Sierpinski's gasket.
The operation takes partial sums of Sierpinski's gasket terms, by columns.
From G. C. Greubel, Dec 02 2024: (Start)
T(n, k) = Sum_{j=k..n} (binomial(j,k) mod 2).
T(n, 0) = A000027(n+1).
T(n, 1) = A004526(n+1).
T(n, 2) = A004524(n+1).
T(2*n, n) = A080100(n).
Sum_{k=0..n} T(n, k) = A006046(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = A006046(floor(n/2)+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A007729(n). (End)

cp's OEIS Frontend

A144337 Triangle read by rows, A000012 * (2*A144328 - A000012).

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A144693 Triangle read by rows, A000012 * (3*A144328 - 2*A000012), where A000012 means a lower triangular matrix of all 1's.

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A145972 Triangle read by rows, A000012 * A053121.

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A152194 Triangle read by rows, A034839 * A000012.

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A153345 Triangle read by rows, A000012 * A055830.

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A153346 Triangle read by rows: A000012 * A153345.

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A157901 Triangle read by rows: A000012 * A157898.

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A158822 Triangle read by rows, matrix triple product A000012 * A145677 * A000012.

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A144693 Triangle read by rows, A000012 * (3A144328 - 2A000012), where A000012 means a lower triangular matrix of all 1's.