A103451 -id:A103451 - cp's OEIS frontend

A135225 Pascal's triangle A007318 augmented with a leftmost border column of 1's.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 4, 1, 1, 1, 5, 10, 10, 5, 1, 1, 1, 6, 15, 20, 15, 6, 1, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums give A094373.
From Peter Bala, Sep 08 2011: (Start)
This augmented Pascal array, call it P, has interesting connections with the Bernoulli polynomials B(n,x). The infinitesimal generator S of P is the array such that exp(S) = P. The array S is obtained by augmenting the infinitesimal generator A132440 of the Pascal triangle with an initial column [0, 0, 1/2, 1/6, 0, -1/30, ...] on the left. The entries in this column, after the first two zeros, are the Bernoulli values B(n,1), n>=1.
The array P is also connected with the problem of summing powers of consecutive integers. In the array P^n, the entry in position p+1 of the first column is equal to sum {k = 1..n} k^p - see the Example section below.
For similar results for the square of Pascal's triangle see A062715.
Note: If we augment Pascal's triangle with the column [1, 1, x, x^2, x^3, ...] on the left, the resulting lower unit triangular array has the Bernoulli polynomials B(n,x) in the first column of its infinitesimal generator. The present case is when x = 1.
(End)

			First few rows of the triangle:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 2, 1;
  1, 1, 3, 3, 1;
  1, 1, 4, 6, 4, 1;
...
The infinitesimal generator for P begins:
  /0
  |0.......0
  |1/2.....1...0
  |1/6.....0...2....0
  |0.......0...0....3....0
  |-1/30...0...0....0....4....0
  |0.......0...0....0....0....5....0
  |1/42....0...0....0....0....0....6....0
  |...
  \
The array P^n begins:
  /1
  |1+1+...+1........1
  |1+2+...+n........n.........1
  |1+2^2+...+n^2....n^2.....2*n........1
  |1+2^3+...+n^3....n^3.....3*n^2....3*n.......1
  |...
  \
More generally, the array P^t, defined as exp(t*S) for complex t, begins:
  /1
  |B(1,1+t)-B(1,1)..........1
  |1/2*(B(2,1+t)-B(2,1))....t.........1
  |1/3*(B(3,1+t)-B(3,1))....t^2.....2*t........1
  |1/4*(B(4,1+t)-B(4,1))....t^3.....3*t^2....3*t.......1
  |...
  \

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

Cf. A007318, A027641, A027642, A062715, A094373, A103438, A132440.

T:= function(n,k)
    if k=0 then return 1;
    else return Binomial(n-1,k-1);
    fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 19 2019

T:= func< n, k | k eq 0 select 1 else Binomial(n-1, k-1) >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 19 2019

T:= proc(n, k) option remember;
      if k=0 then 1
    else binomial(n-1, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 19 2019

T[n_, k_]:= T[n, k]= If[k==0, 1, Binomial[n-1, k-1]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 19 2019 *)

T(n,k) = if(k==0, 1, binomial(n-1, k-1)); \\ G. C. Greubel, Nov 19 2019

@CachedFunction
def T(n,k):
    if (k==0): return 1
    else: return binomial(n-1, k-1)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 19 2019

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

Given A007318, binomial(n,k) is shifted to T(n+1,k+1) and a leftmost border of 1's is added.

Corrected by R. J. Mathar, Apr 16 2013

A154325 Triangle with interior all 2's and borders 1.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Jan 07 2009

This triangle follows a general construction method as follows: Let a(n) be an integer sequence with a(0)=1, a(1)=1. Then T(n,k,r):=[k<=n](1+r*a(k)*a(n-k)) defines a symmetrical triangle.
Row sums are n + 1 + r*Sum_{k=0..n} a(k)*a(n-k) and central coefficients are 1+r*a(n)^2.
Here a(n)=1-0^n and r=1. Row sums are A004277.
Eigensequence of the triangle = A000129, the Pell sequence. - Gary W. Adamson, Feb 12 2009
Inverse has general element T(n,k)*(-1)^(n-k). - Paul Barry, Oct 06 2010

			Triangle begins
  1;
  1, 1;
  1, 2, 1;
  1, 2, 2, 1;
  1, 2, 2, 2, 1;
  1, 2, 2, 2, 2, 1;
  1, 2, 2, 2, 2, 2, 1;
From _Paul Barry_, Oct 06 2010: (Start)
Production matrix is
  1,  1;
  0,  1, 1;
  0, -1, 0, 1;
  0,  1, 0, 0, 1;
  0, -1, 0, 0, 0, 1;
  0,  1, 0, 0, 0, 0, 1;
  0, -1, 0, 0, 0, 0, 0, 1;
  0,  1, 0, 0, 0, 0, 0, 0, 1; (End)

Cf. A000129, A103451, A129765.

a[n_] :=
 If[Length@
    NestWhileList[# -
       Floor[(Sqrt[8 # + 1] - 1)/2] (Floor[(Sqrt[8 # + 1] - 1)/2] + 1)/
2 &, n, # > 1 &] <= 2, 1, 2] (* David Naccache, Jul 13 2025 *)

row(n) = vector(n+1, k, k--; (2-0^(k*(n-k)))); \\ Michel Marcus, Jul 13 2025

Number triangle T(n,k) = [k<=n](2-0^(n-k)-0^k+0^(n+k)) = [k<=n](2-0^(k*(n-k))).

a(n) = 2 - A103451(n). - Omar E. Pol, Jan 18 2009

More terms from Michel Marcus, Jul 13 2025

A132735 Triangle T(n,k) = binomial(n,k) + 1 with T(n,0) = T(n,n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 7, 5, 1, 1, 6, 11, 11, 6, 1, 1, 7, 16, 21, 16, 7, 1, 1, 8, 22, 36, 36, 22, 8, 1, 1, 9, 29, 57, 71, 57, 29, 9, 1, 1, 10, 37, 85, 127, 127, 85, 37, 10, 1, 1, 11, 46, 121, 211, 253, 211, 121, 46, 11, 1, 1, 12, 56, 166, 331, 463, 463, 331, 166, 56, 12, 1

Offset: 0

Gary W. Adamson, Aug 26 2007

			First few rows of the triangle are:
  1;
  1, 1;
  1, 3,  1;
  1, 4,  4,  1;
  1, 5,  7,  5,  1;
  1, 6, 11, 11,  6, 1;
  1, 7, 16, 21, 16, 7, 1;
  ...

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

Cf. A103451, A132736.

Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), this sequence (q=1), A173740 (q=2), A173741 (q=4), A173742 (q=6).

T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) + 1 >;
[T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel, Feb 14 2021

T[n_, k_]:= If[k==0||k==n, 1, Binomial[n,k] +1];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 14 2021 *)

def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 1
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 14 2021

T(n, k) = A007318(n,k) + 1 - A103451(n,k), an infinite lower triangular matrix.
T(n,0) = T(n,n) = 1; T(n,k) = C(n,k) + 1 otherwise. - Franklin T. Adams-Watters, Jul 06 2009
Sum_{k=0..n} T(n, k) = 2^n + n - 1 + [n=0] = A132736(n). - G. C. Greubel, Feb 14 2021

Corrected and extended by Franklin T. Adams-Watters, Jul 06 2009

A173740 Triangle T(n,k) = binomial(n,k) + 2 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 6, 8, 6, 1, 1, 7, 12, 12, 7, 1, 1, 8, 17, 22, 17, 8, 1, 1, 9, 23, 37, 37, 23, 9, 1, 1, 10, 30, 58, 72, 58, 30, 10, 1, 1, 11, 38, 86, 128, 128, 86, 38, 11, 1, 1, 12, 47, 122, 212, 254, 212, 122, 47, 12, 1, 1, 13, 57, 167, 332, 464, 464, 332, 167, 57, 13, 1

Offset: 0

Roger L. Bagula, Feb 23 2010

For n >= 1, row n sums to A131520(n).

			Triangle begins:
  1;
  1,  1;
  1,  4,  1;
  1,  5,  5,   1;
  1,  6,  8,   6,   1;
  1,  7, 12,  12,   7,   1;
  1,  8, 17,  22,  17,   8,   1;
  1,  9, 23,  37,  37,  23,   9,   1;
  1, 10, 30,  58,  72,  58,  30,  10,  1;
  1, 11, 38,  86, 128, 128,  86,  38, 11,  1;
  1, 12, 47, 122, 212, 254, 212, 122, 47, 12, 1;
  ...

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

Cf. A100314, A103451, A131520, A156050.

Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), A132735 (q=1), this sequence (q=2), A173741 (q=4), A173742 (q=6).

T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) + 2 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021

T[n_, m_] = Binomial[n, m] + 2*If[m*(n - m) > 0, 1, 0];
Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]

T(n,k) := if k = 0 or k = n then 1 else binomial(n, k) + 2$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 08 2018 */

def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 2
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

From Franck Maminirina Ramaharo, Dec 08 2018:(Start)
T(n,k) = A007318(n,k) + 2*(1 - A103451(n,k)).
T(n,k) = 3*A007318(n,k) - 2*A132044(n,k).
n-th row polynomial is 1 - (-1)^(2^n) + (1 + x)^n + 2*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 3*x*y^2 - 2*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (2 - 2*x + 2*x*exp(y) - 2*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 2*(n - 1 + [n=0]) = 2*A100314(n). - G. C. Greubel, Feb 13 2021

Edited and name clarified by Franck Maminirina Ramaharo, Dec 08 2018

A132731 Triangle T(n,k) = 2 * binomial(n,k) - 2 with T(n,0) = T(n,n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 10, 6, 1, 1, 8, 18, 18, 8, 1, 1, 10, 28, 38, 28, 10, 1, 1, 12, 40, 68, 68, 40, 12, 1, 1, 14, 54, 110, 138, 110, 54, 14, 1, 1, 16, 70, 166, 250, 250, 166, 70, 16, 1, 1, 18, 88, 238, 418, 502, 418, 238, 88, 18, 1

Offset: 0

Gary W. Adamson, Aug 26 2007

			First few rows of the triangle are:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  4,  1;
  1,  6, 10,  6,  1;
  1,  8, 18, 18,  8,  1;
  1, 10, 28, 38, 28, 10,  1;
  1, 12, 40, 68, 68, 40, 12, 1;
  ...

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

Cf. A000012, A007318, A103451, A132044, A132732 (row sums).

T:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) - 2 >;
[T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel, Feb 14 2021

T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n, k] - 2];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 14 2021 *)

t(n,k) =  2*binomial(n, k) + ((k==0) || (k==n)) - 2*(k<=n); \\ Michel Marcus, Feb 12 2014

def T(n, k): return 1 if (k==0 or k==n) else 2*binomial(n, k) - 2
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 14 2021

T(n, k) = 2*A007318 + A103451 - 2*A000012, an infinite lower triangular matrix.
From G. C. Greubel, Feb 14 2021: (Start)
T(n, k) = 2*binomial(n, k) - 2 with T(n, 0) = T(n, n) = 1.
T(n, k) = 2*A132044(n, k) with T(n, 0) = T(n, n) = 1.
Sum_{k=0..n} T(n, k) = 2^(n+1) - 2*n - [n=0] = A132732(n). (End)

Corrected by Jeremy Gardiner, Feb 02 2014

More terms from Michel Marcus, Feb 12 2014

A132749 Triangle T(n,k) = binomial(n, k) with T(n, 0) = 2, read by rows.

Original entry on oeis.org

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

Offset: 0

Gary W. Adamson, Aug 28 2007

Add 1 to all but the top entry in the left column of the Pascal matrix. - R. J. Mathar, Jan 18 2013

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

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

Cf. A007318, A083318 (row sums), A103451.

A132749:= func< n,k | k eq n select 1 else k eq 0 select 2 else Binomial(n,k) >;
[A132749(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

T[n_, k_]:= If[k==n, 1, If[k==0, 2, Binomial[n, k]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 16 2021 *)

def A132749(n,k): return 1 if k==n else 2 if k==0 else binomial(n,k)
flatten([[A132749(n,k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Feb 16 2021

T(n,k) = A103451(n,k) * A007318(n,k), an infinite lower triangular matrix.
From G. C. Greubel, Feb 16 2021: (Start)
T(n,k) = binomial(n, k) with T(n, 0) = 2 for n>0.
Sum_{k=0..n} T(n, k) = A083318(n) = 2^n + 1^n - 0^n. (End)

More terms added by G. C. Greubel, Feb 16 2021

A173741 T(n,k) = binomial(n,k) + 4 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 7, 7, 1, 1, 8, 10, 8, 1, 1, 9, 14, 14, 9, 1, 1, 10, 19, 24, 19, 10, 1, 1, 11, 25, 39, 39, 25, 11, 1, 1, 12, 32, 60, 74, 60, 32, 12, 1, 1, 13, 40, 88, 130, 130, 88, 40, 13, 1, 1, 14, 49, 124, 214, 256, 214, 124, 49, 14, 1, 1, 15, 59, 169, 334, 466, 466, 334, 169, 59, 15, 1

Offset: 0

Roger L. Bagula, Feb 23 2010

For n >= 1, row n sums to 2*A100314(n).

			Triangle begins:
  1;
  1,  1;
  1,  6,  1;
  1,  7,  7,   1;
  1,  8, 10,   8,   1;
  1,  9, 14,  14,   9,   1;
  1, 10, 19,  24,  19,  10,   1;
  1, 11, 25,  39,  39,  25,  11,   1;
  1, 12, 32,  60,  74,  60,  32,  12,  1;
  1, 13, 40,  88, 130, 130,  88,  40, 13,  1;
  1, 14, 49, 124, 214, 256, 214, 124, 49, 14, 1;
  ...

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

Cf. A100314, A103451, A156050.

Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), A132735 (q=1), A173740 (q=2), this sequence (q=4), A173742 (q=6).

T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) + 4 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021

T[n_, m_] = Binomial[n, m] + 4*If[m*(n - m) > 0, 1, 0];
Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]

T(n,k) := if k = 0 or k = n then 1 else binomial(n, k) + 4$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 09 2018 */

def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 4
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

From Franck Maminirina Ramaharo, Dec 09 2018:(Start)
T(n,k) = A007318(n,k) + 2*(1 - A103451(n,k)).
T(n,k) = 5*A007318(n,k) - 4*A132044(n,k).
n-th row polynomial is 2*(1 - (-1)^(2^n)) + (1 + x)^n + 4*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 5*x*y^2 - 4*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (4 - 4*x + 4*x*exp(y) - 4*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 4*(n - 1 + [n=0]) = 2*A100314(n). - G. C. Greubel, Feb 13 2021

Edited and name clarified by Franck Maminirina Ramaharo, Dec 09 2018

A173742 Triangle T(n,k) = binomial(n,k) + 6 with T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 9, 9, 1, 1, 10, 12, 10, 1, 1, 11, 16, 16, 11, 1, 1, 12, 21, 26, 21, 12, 1, 1, 13, 27, 41, 41, 27, 13, 1, 1, 14, 34, 62, 76, 62, 34, 14, 1, 1, 15, 42, 90, 132, 132, 90, 42, 15, 1, 1, 16, 51, 126, 216, 258, 216, 126, 51, 16, 1, 1, 17, 61, 171, 336, 468, 468, 336, 171, 61, 17, 1

Offset: 0

Roger L. Bagula, Feb 23 2010

For n >= 1, row n sums to A131520(n) + A008586(n).

			Triangle begins:
  1;
  1,  1;
  1,  8,  1;
  1,  9,  9,   1;
  1, 10, 12,  10,   1;
  1, 11, 16,  16,  11,   1;
  1, 12, 21,  26,  21,  12,   1;
  1, 13, 27,  41,  41,  27,  13,   1;
  1, 14, 34,  62,  76,  62,  34,  14,  1;
  1, 15, 42,  90, 132, 132,  90,  42, 15,  1;
  1, 16, 51, 126, 216, 258, 216, 126, 51, 16, 1;
  ...

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

Cf. A007318, A103451, A132044, A156050, A173740, A173741.

T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) +6 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021

T[n_, m_] = Binomial[n, m] + 6*If[m*(n - m) > 0, 1, 0];
Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]

T(n,k) := if k = 0 or k = n then 1 else binomial(n, k) + 6$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 09 2018 */

def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 6
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

From Franck Maminirina Ramaharo, Dec 09 2018: (Start)
T(n,k) = A007318(n,k) + 6*(1 - A103451(n,k)).
T(n,k) = 7*A007318(n,k) - 6*A132044(n,k).
n-th row polynomial is 3*(1 - (-1)^(2^n)) + (1 + x)^n + 6*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 7*x*y^2 - 6*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (6 - 6*x + 6*x*exp(y) - 6*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 6*n - 6 + 6*[n=0]. - G. C. Greubel, Feb 13 2021

Edited and name clarified by Franck Maminirina Ramaharo, Dec 09 2018

A185784 Accumulation array of A107985, by antidiagonals.

Original entry on oeis.org

1, 4, 4, 10, 15, 10, 20, 36, 36, 20, 35, 70, 84, 70, 35, 56, 120, 160, 160, 120, 56, 84, 189, 270, 300, 270, 189, 84, 120, 280, 420, 500, 500, 420, 280, 120, 165, 396, 616, 770, 825, 770, 616, 396, 165, 220, 540, 864, 1120, 1260, 1260, 1120, 864, 540, 220, 286, 715, 1170, 1560, 1820, 1911, 1820, 1560, 1170, 715, 286, 364, 924, 1540, 2100, 2520, 2744, 2744, 2520, 2100, 1540, 924, 364, 455, 1170, 1980, 2750, 3375, 3780, 3920, 3780, 3375, 2750, 1980, 1170, 455, 560

Offset: 1

Clark Kimberling, Feb 03 2011

Let W be the array given by w(1,1)=1, w(2,2)=-1, and w(n,k)=0 for all other (n,k).

Write "A < B" to indicate that an array B is the accumulation array of A (defined at A144112). Then W < A103451 < A002024 < A107985 < A185784 < A185785 < A185786.

			Northwest corner:
1....4....10....20....35
4....15...36....70....120
10...36...84....160...270
20...70...160...300...500

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

Cf. A144112, A103451, A002024, A107985, A185784, A185785, A185786.

(* The code generates arrays A107985, A185784, A002024. *)
f[n_,0]:=0;f[0,k_]:=0; (* used to form A002024 *)
f[n_,k_]:=k*n(k+n)/2;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A107985 *)
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]]
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* A185784 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A002024 *)

T(n,k) = (n+k+1)*C(n+1,2)*C(k+1,2)/3, k>=0, n>=0.

A091918 Inverse of number triangle A091917.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 3, 3, 1, 16, 4, 6, 4, 1, 32, 5, 10, 10, 5, 1, 64, 6, 15, 20, 15, 6, 1, 128, 7, 21, 35, 35, 21, 7, 1, 256, 8, 28, 56, 70, 56, 28, 8, 1, 512, 9, 36, 84, 126, 126, 84, 36, 9, 1, 1024, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 2048, 11, 55, 165, 330, 462, 462, 330

Offset: 0

Paul Barry, Feb 13 2004

Essentially, Pascal's triangle A007318 with first column of 1's replaced by 2^n Row sums are A000225(n+1). Diagonal sums are A000225(n) + A000045(n).

Cf. A103451.

Binomial transform of triangle A103451: (1; 1,1; 1,0,1; 1,0,0,1; ...). - Gary W. Adamson, Aug 08 2007

cp's OEIS Frontend

A135225 Pascal's triangle A007318 augmented with a leftmost border column of 1's.

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A154325 Triangle with interior all 2's and borders 1.

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A132735 Triangle T(n,k) = binomial(n,k) + 1 with T(n,0) = T(n,n) = 1, read by rows.

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A173740 Triangle T(n,k) = binomial(n,k) + 2 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

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A132731 Triangle T(n,k) = 2 * binomial(n,k) - 2 with T(n,0) = T(n,n) = 1, read by rows.

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A132749 Triangle T(n,k) = binomial(n, k) with T(n, 0) = 2, read by rows.

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