A014430 -id:A014430 - cp's OEIS frontend

A129696 Antidiagonal sums of triangular array T defined in A014430: T(j,k) = binomial(j+1, k) - 1 for 1 <= k <= j.

1, 2, 5, 9, 17, 29, 50, 83, 138, 226, 370, 602, 979, 1588, 2575, 4171, 6755, 10935, 17700, 28645, 46356, 75012, 121380, 196404, 317797, 514214, 832025, 1346253, 2178293, 3524561, 5702870, 9227447, 14930334, 24157798, 39088150, 63245966

Offset: 1

Paul Curtz, Jun 01 2007

If T is construed as a lower triangular matrix M over the rational field, the inverse M^-1 is a lower triangular matrix containing fractions. Its row sums are the Bernoulli numbers. First column of M^-1 is 1, -1, 2/3, -1/4, -1/30, 1/12, 1/42, -1/12, ... . Multiplied by j! this gives 1, -2, 4, -6, -4, 60, 120, -3660, ... .
The Kn22 sums, see A180662 for the definition of these sums, of the 'Races with Ties' triangle A035317 lead to this sequence. - Johannes W. Meijer, Jul 20 2011
This sequence is the convolution of (1,1,2,3,5,8,13,21,...) and (1,1,2,2,3,3,4,4,5,5,...), i.e., the Fibonacci numbers (A000045) and A008619. - Clark Kimberling, May 28 2012
a(n) is the sum of the first summands over all Arndt compositions of n (see the Checa link). - Daniel Checa, Jan 01 2024

Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, 1969.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Daniel F. Checa, Arndt Compositions: Connections with Fibonacci Numbers, Statistics, and Generalizations, 2023, p. 31.
Larry Ericksen, Problem 7, Problem Section, The Fibonacci Quarterly, Vol. 57, No. 5 (2019), pp. 172, 175-183.
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).

Cf. A000045, A000071, A001924, A008619, A014430, A027641, A035317, A052952 (first differences), A180662.

Cf. A000120, A002487.

m:=36; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=1 to j do M[j, k]:=Binomial(j+1, k)-1; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // Klaus Brockhaus, Jun 11 2007

[Fibonacci(n+3)-2-Floor(n/2): n in [1..40]]; // Vincenzo Librandi, Nov 23 2014

with(combinat): a := proc (n) options operator, arrow: fibonacci(n+3)-2-floor((1/2)*n) end proc: seq(a(n), n = 1 .. 34); # Emeric Deutsch, Nov 22 2014

a[n_]:= a[n]= If[n<3, n, a[n-1] + a[n-2] + (n + Mod[n, 2])/2];
Table[a[n], {n,40}] (* Jean-François Alcover, Mar 04 2013 *)
Table[Fibonacci[n+3] -2 -Floor[n/2], {n, 100}] (* Vincenzo Librandi, Nov 23 2014 *)

prpr = 1
prev = 2
for n in range(2,100):
    print(prpr, end=", ")
    curr = prpr+prev + 1 + n//2
    prpr = prev
    prev = curr
# Alex Ratushnyak, Jul 30 2012

[fibonacci(n+3) -2 -(n//2) for n in range(1,41)] # G. C. Greubel, Mar 17 2023

From Paul Barry, Jan 18 2009: (Start)
a(n) = Sum_{k=0..floor(n/2)} A000071(n-2*k+3).
a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-2*k} Fibonacci(j+1)). (End)
a(n+1) = a(n-1) + a(n) + 1 + floor(n/2) for n>1, a(0)=1, a(1)=2. - Alex Ratushnyak, Jul 30 2012
From R. J. Mathar, Jul 25 2013: (Start)
G.f.: x/((1 + x)*(1 - x)^2*(1 - x - x^2)).
a(n) + a(n+1) = A001924(n+1). (End)
a(n) = Fibonacci(n+3) - 2 - floor(n/2). - Emeric Deutsch, Nov 22 2014
a(n) = (-5/4 - (-1)^n/4 + (2^(-n)*((1 - t)^n*(-2 + t) + (1 + t)^n*(2 + t)))/t + (-1 - n)/2), where t=sqrt(5). - Colin Barker, Feb 09 2017
E.g.f.: (4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2)) - 5*(4 + x)*cosh(x) - 5*(3 + x)*sinh(x))/10. - Stefano Spezia, Apr 06 2024
a(n) = max_{k = 2^(n+1)..2^(n+2)-1} (A002487(k) - A000120(k)) (Ericksen, 2019). - Amiram Eldar, Jan 30 2025

Edited and extended by Klaus Brockhaus, Jun 11 2007

A014473 Pascal's triangle - 1: Triangle read by rows: T(n, k) = A007318(n, k) - 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 9, 9, 4, 0, 0, 5, 14, 19, 14, 5, 0, 0, 6, 20, 34, 34, 20, 6, 0, 0, 7, 27, 55, 69, 55, 27, 7, 0, 0, 8, 35, 83, 125, 125, 83, 35, 8, 0, 0, 9, 44, 119, 209, 251, 209, 119, 44, 9, 0, 0, 10, 54, 164, 329, 461, 461, 329, 164, 54, 10, 0

Offset: 0

N. J. A. Sloane

Indexed as a square array A(n,k): If X is an (n+k)-set and Y a fixed k-subset of X then A(n,k) is equal to the number of n-subsets of X intersecting Y. - Peter Luschny, Apr 20 2012

			Triangle begins:
   0;
   0, 0;
   0, 1,  0;
   0, 2,  2,  0;
   0, 3,  5,  3,  0;
   0, 4,  9,  9,  4,  0;
   0, 5, 14, 19, 14,  5, 0;
   0, 6, 20, 34, 34, 20, 6, 0;
   ...
Seen as a square array read by antidiagonals:
  [0] 0, 0,  0,  0,   0,   0,   0,    0,    0,    0,    0,     0, ... A000004
  [1] 0, 1,  2,  3,   4,   5,   6,    7,    8,    9,   10,    11, ... A001477
  [2] 0, 2,  5,  9,  14,  20,  27,   35,   44,   54,   65,    77, ... A000096
  [3] 0, 3,  9, 19,  34,  55,  83,  119,  164,  219,  285,   363, ... A062748
  [4] 0, 4, 14, 34,  69, 125, 209,  329,  494,  714, 1000,  1364, ... A063258
  [5] 0, 5, 20, 55, 125, 251, 461,  791, 1286, 2001, 3002,  4367, ... A062988
  [6] 0, 6, 27, 83, 209, 461, 923, 1715, 3002, 5004, 8007, 12375, ... A124089

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
Milan Janjic, Two Enumerative Functions

Triangle without zeros: A014430.
Related: A323211 (A007318(n, k) + 1).
A000295 (row sums), A059841 (alternating row sums), A030662(n-1) (central terms).
Columns include A000096, A062748, A062988, A063258.
Diagonals of A(n, n+d): A030662 (d=0), A010763 (d=1), A322938 (d=2).
Cf. A000004, A001477, A007318, A030662, A059841, A109128, A124089, A129696.

a014473 n k = a014473_tabl !! n !! k
a014473_row n = a014473_tabl !! n
a014473_tabl = map (map (subtract 1)) a007318_tabl
-- Reinhard Zumkeller, Apr 10 2012

[Binomial(n,k)-1: k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 08 2024

with(combstruct): for n from 0 to 11 do seq(-1+count(Combination(n), size=m), m = 0 .. n) od; # Zerinvary Lajos, Apr 09 2008
# The rows of the square array:
Arow := proc(n, len) local gf, ser;
gf := (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1));
ser := series(gf, x, len+2): seq((-1)^(n+1)*coeff(ser, x, j), j=0..len) end:
for n from 0 to 9 do lprint([n], Arow(n, 12)) od; # Peter Luschny, Feb 13 2019

Table[Binomial[n,k] -1, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 08 2024 *)

flatten([[binomial(n,k)-1 for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 08 2024

G.f.: x^2*y/((1 - x)*(1 - x*y)*(1 - x*(1 + y))). - Ralf Stephan, Jan 24 2005
T(n,k) = A109128(n,k) - A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Apr 10 2012
T(n, k) = T(n-1, k-1) + T(n-1, k) + 1, 0 < k < n with T(n, 0) = T(n, n) = 0. - Reinhard Zumkeller, Jul 18 2015
If seen as a square array read by antidiagonals the generating function of row n is: G(n) = (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1)). - Peter Luschny, Feb 13 2019
From G. C. Greubel, Apr 08 2024: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..floor(n/2)} T(n-k, k) = A129696(n-2).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = b(n-1), where b(n) is the repeating pattern {0, 0, -1, -2, -1, 1, 1, -1, -2, -1, 0, 0}_{n=0..11}, with b(n) = b(n-12). (End)

More terms from Erich Friedman

A185778 Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals.

Original entry on oeis.org

1, -1, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 0, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 0, 0, 0, 1, 8, 28, 56, 70, 56, 28, 8, 1, 0, 0, 0, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 0

Offset: 1

Clark Kimberling, Feb 03 2011

Using "->" to mean "is the weight array of" as defined at A144112:

A185779->A144225->A007318->A014430->A077023->A185779, where each of these is formatted as a rectangle (e.g., A007318 is Pascal's triangle). Read in reverse order, each is the accumulation array of the preceding array. It appears that successive weight arrays of A185779 contain Pascal's triangle except for initial terms.

			Northwest corner:
1....-1....0....0....0....0....0,...0
-1....2....0....0....0....0....0....0
0.....0....0....1....1....1....1....1
0.....0....1....2....3....4....5....6
0.....0....1....3....6....10...15...21
0.....0....1....4....10...20...35...56

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

Cf. A144112, A185779, A144225, A007318, A014430, A077023, A185779.

(* This code produces three arrays: A144225, A007318, A185778. *)
f[n_,0]:=0;f[0,k_]:=0;  (* Used to make the weight array *)
f[1,1]:=1;f[n_,1]:=0;f[1,k_]:=0
f[n_,2]:=1;f[2,k_]:=1;
f[n_,k_]:=-1+(n+k-4)!/((n-2)!*(k-2)!)/;k>1&&n>1;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A144225 *)
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A007318, Pascal's triangle formatted as a rectangle *)
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}]] (* A185778 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

(See the Mathematica code.)

A337787 Number of addition triangles whose sum is n (version 2).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 6, 3, 5, 2, 8, 2, 8, 4, 8, 3, 12, 3, 11, 6, 11, 5, 15, 4, 16, 9, 14, 7, 20, 8, 18, 11, 20, 12, 25, 8, 25, 18, 24, 12, 31, 16, 32, 19, 29, 21, 39, 19, 36, 28, 38, 25, 47, 25, 46, 33, 46, 34, 55, 31, 56, 44, 55, 39, 67, 42, 66, 52, 66, 53, 76, 50, 81, 65, 77, 57

Offset: 1

Seiichi Manyama, Sep 21 2020

An addition triangle has any set of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.

Reversing the base does not count as a different triangle.

			   n |
-----+-------------------------------
   1 |  1
-----+-------------------------------
   2 |  2
-----+-------------------------------
   3 |  3
-----+-------------------------------
   4 |      2
     |  4  1,1
-----+-------------------------------
   5 |  5
-----+-------------------------------
   6 |      3
     |  6  1,2
-----+-------------------------------
   7 |  7
-----+-------------------------------
   8 |      4    4
     |  8  1,3  2,2
-----+-------------------------------
   9 |  9
-----+-------------------------------
  10 |      5    5
     | 10  1,4  2,3
-----+-------------------------------
  11 |       4
     |      2,2
     | 11  1,1,1
-----+-------------------------------
  12 |      6    6    6
     | 12  1,5  2,4  3,3
-----+-------------------------------
  13 | 13
-----+-------------------------------
  14 |                      5
     |      7    7    7    2,3
     | 14  1,6  2,5  3,4  1,1,2
-----+-------------------------------
  15 | 15
-----+-------------------------------
  16 |                           6
     |      8    8    8    8    3,3
     | 16  1,7  2,6  3,5  4,4  1,2,1
-----+-------------------------------
  17 |       6      6
     |      2,4    3,3
     | 17  1,1,3  2,1,2
-----+-------------------------------
  18 |      9    9    9    9
     | 18  1,8  2,7  3,6  4,5
-----+-------------------------------
  19 |       7
     |      3,4
     | 19  1,2,2

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A014430, A062684, A062896, see A337785 for version 1.

def f(n)
  ary = [1]
  (n - 1).times{|i|
    ary = [0] + ary + [0]
    ary = (0..i + 1).map{|j| ary[j] + ary[j + 1] + 1}
  }
  ary
end
def A(n)
  f_ary = (1..n / 2).map{|i| [i]}
  cnt = 2
  s = 1
  while f_ary.size > 0
    s_ary = f(s + 1)
    b_ary = []
    f_ary.each{|i|
      (1..i[0] - 1).each{|j|
        a = [j]
        (0..s - 1).each{|k|
          num = i[k] - a[k]
          if num > 0
            a << num
          else
            break
          end
        }
        if a.size == s + 1
          sum = (0..s).inject(0){|t, m| t + s_ary[m] * a[m]}
          if sum < n
            b_ary << a
          elsif sum == n
            cnt += 1
            cnt += 1 if a == a.reverse
          end
        end
      }
    }
    f_ary = b_ary
    s += 1
  end
  cnt / 2
end
def A337787(n)
  (1..n).map{|i| A(i)}
end
p A337787(50)

A185779 Third accumulation array of Pascal's triangle (as a rectangle), by antidiagonals.

Original entry on oeis.org

1, 4, 4, 10, 17, 10, 20, 45, 45, 20, 35, 95, 126, 95, 35, 56, 175, 281, 281, 175, 56, 84, 294, 546, 662, 546, 294, 84, 120, 462, 966, 1358, 1358, 966, 462, 120, 165, 690, 1596, 2534, 2941, 2534, 1596, 690, 165, 220, 990, 2502, 4410, 5790, 5790, 4410, 2502, 990, 220, 286, 1375, 3762, 7272, 10620, 12021, 10620, 7272, 3762, 1375, 286, 364, 1859, 5467, 11484, 18432, 23229, 23229, 18432, 11484, 5467, 1859, 364, 455

Offset: 1

Clark Kimberling, Feb 03 2011

Using "Axxxxxx < Ayyyyyy" to mean that Ayyyyyy is the accumulation array of Axxxxxx, as defined at A144112:
A185779 < A144225 < A007318 < A014430 < A077023 < A185779, where each of these is formatted as a rectangle (e.g., A007318 is Pascal's triangle).  See A185778.
row 1: A000292
row 2: A095667

			Northwest corner:
1....4...10...20...35
4....17..45...95...175
10...45..126..281..546
20...95..281..662..1358

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

Cf. A007318, A014430, A077023, A144112, A144225, A185778.

f[n_, k_] := Binomial[n + k + 4, n + 2] - (k + 3)*(k + 4)/2 - (k + 2)* n*(k*n + n + 3*k + 7)/4; TableForm[Table[f[n, k], {n, 1, 5}, {k, 1, 5}]]
Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 12 2017 *)

T(n,k) = C(n+k+4,n+2) - (k+3)*(k+4)/2 - (k+2)*n*(k*n+n+3*k+7)/4, for k>=1, n>=1.

A337785 Number of addition triangles whose sum is n (version 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 1, 9, 1, 9, 4, 9, 3, 14, 2, 14, 6, 14, 5, 21, 4, 19, 10, 21, 8, 27, 6, 29, 16, 25, 12, 38, 14, 33, 19, 37, 22, 46, 14, 47, 33, 45, 22, 59, 29, 59, 35, 56, 40, 74, 34, 68, 53, 72, 47, 90, 47, 88, 63, 88, 64, 105, 59, 108, 84, 106, 75, 130, 81, 125, 99, 128, 103, 147

Offset: 1

Seiichi Manyama, Sep 21 2020

An addition triangle has any set of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.

Reversing the base counts as a different triangle.

			   n |
-----+------------------------------------------------
   1 |  1
-----+------------------------------------------------
   2 |  2
-----+------------------------------------------------
   3 |  3
-----+------------------------------------------------
   4 |      2
     |  4  1,1
-----+------------------------------------------------
   5 |  5
-----+------------------------------------------------
   6 |      3    3
     |  6  1,2  2,1
-----+------------------------------------------------
   7 |  7
-----+------------------------------------------------
   8 |      4    4    4
     |  8  1,3  2,2  3,1
-----+------------------------------------------------
   9 |  9
-----+------------------------------------------------
  10 |      5    5    5    5
     | 10  1,4  2,3  3,2  4,1
-----+------------------------------------------------
  11 |       4
     |      2,2
     | 11  1,1,1
-----+------------------------------------------------
  12 |      6    6    6    6    6
     | 12  1,5  2,4  3,3  4,2  5,1
-----+------------------------------------------------
  13 | 13
-----+------------------------------------------------
  14 |                                     5      5
     |      7    7    7    7    7    7    2,3    3,2
     | 14  1,6  2,5  3,4  4,3  5,2  6,1  1,1,2  2,1,1

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A014430, A062684, A062896, A337765, A337766, see A337787 for version 2.

def f(n)
  ary = [1]
  (n - 1).times{|i|
    ary = [0] + ary + [0]
    ary = (0..i + 1).map{|j| ary[j] + ary[j + 1] + 1}
  }
  ary
end
def A(n)
  f_ary = (1..n / 2).map{|i| [i]}
  cnt = 1
  s = 1
  while f_ary.size > 0
    s_ary = f(s + 1)
    b_ary = []
    f_ary.each{|i|
      (1..i[0] - 1).each{|j|
        a = [j]
        (0..s - 1).each{|k|
          num = i[k] - a[k]
          if num > 0
            a << num
          else
            break
          end
        }
        if a.size == s + 1
          sum = (0..s).inject(0){|t, m| t + s_ary[m] * a[m]}
          if sum < n
            b_ary << a
          elsif sum == n
            cnt += 1
          end
        end
      }
    }
    f_ary = b_ary
    s += 1
  end
  cnt
end
def A337785(n)
  (1..n).map{|i| A(i)}
end
p A337785(50)

A124326 T(n,m) = A007318(n,m) - A077028(n,m).

Original entry on oeis.org

1, 3, 3, 6, 10, 6, 10, 22, 22, 10, 15, 40, 53, 40, 15, 21, 65, 105, 105, 65, 21, 28, 98, 185, 226, 185, 98, 28, 36, 140, 301, 431, 431, 301, 140, 36, 45, 192, 462, 756, 887, 756, 462, 192, 45, 55, 255, 678, 1246, 1673, 1673, 1246, 678, 255, 55, 66, 330, 960, 1956, 2954

Offset: 1

Roger L. Bagula, Jun 26 2007

First term of n-th row is n*(n+1)/2.
Row sum are A002663 (without initial zeros).
Appears to be the triangle resulting from adding the row number (first row numbered 0) of Pascal's triangle (A007318) to each entry in that row, subtracting the corresponding entries in the triangle formed by taking the finite diagonals in the multiplication table in order of increasing length (A003991), and removing the outer two layers, which consist entirely of 0's.
Each value of the sequence T(x,y) is equal to the sum of all values in A014430 that are in the rectangle defined by the tip (0,0) and the position (x,y). - Jon Perry, Sep 11 2013

			Table begins
   1;
   3,   3;
   6,  10,   6;
  10,  22,  22,  10;
  15,  40,  53,  40,  15;
  21,  65, 105, 105,  65,  21;
  28,  98, 185, 226, 185,  98,  28;
  36, 140, 301, 431, 431, 301, 140,  36;
  45, 192, 462, 756, 887, 756, 462, 192,  45;
  ...
If the zeros are included, the table begins
   0;
   0, 0;
   0, 0, 0;
   0, 0, 0, 0;
   0, 0, 1, 0, 0;
   0, 0, 3, 3, 0, 0;
   0, 0, 6, 10, 6, 0, 0;
   0, 0, 10, 22, 22, 10, 0, 0;
   ... - from _Michael De Vlieger_, May 27 2024

Iva Kodrnja and Helena Koncul, Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N)), arXiv:2405.10747 [math.NT], 2024. See p. 10, also Glasnik Matematički, (2024) Vol. 59, No. 79, 313-325. See pp. 319-320.

Cf. A077028, A007318, A002663, A014430.

a = Table[Flatten[Table[If[Binomial[m, n] - (1 +n (m - n)) == 0, {}, Binomial[m, n] - (1 + n (m - n))], {n, 0, m}]], {m, 0, 14}]

T(n,m) = A007318(n,m) - A077028(n,m) (skipping zeros).

A050174 T(n,k) = S(n,k,k-2), 1<=k<=n-2, n >= 3, array S as in A050157.

Original entry on oeis.org

1, 2, 2, 3, 5, 5, 4, 9, 14, 9, 5, 14, 28, 28, 14, 6, 20, 48, 62, 48, 20, 7, 27, 75, 117, 117, 75, 27, 8, 35, 110, 200, 242, 200, 110, 35, 9, 44, 154, 319, 451, 451, 319, 154, 44, 10, 54, 208, 483, 780, 912, 780, 483, 208, 54, 11, 65, 273, 702, 1274

Offset: 3

Clark Kimberling

S(n,k,k) = C(n,k) for 1<=k<=n, n >= 1; cf. Pascal's triangle, A007318.

S(n,k,k-1), 1<=k<=n-1, n >= 2 is given by A014430.

			Rows:
1;
2, 2;
3, 5, 5;
4, 9, 14, 9;
5, 14, 28, 28, 14;
...

Cf. A007318, A014430.

T(n,k) = C(n,k) - n for k>1, and T(n, 1) = n - 2. - Andrei Asinowski, Jan 27 2016

Offset changed to 3 by Michel Marcus, Jan 29 2016

cp's OEIS Frontend

A129696 Antidiagonal sums of triangular array T defined in A014430: T(j,k) = binomial(j+1, k) - 1 for 1 <= k <= j.

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A014473 Pascal's triangle - 1: Triangle read by rows: T(n, k) = A007318(n, k) - 1.

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A185778 Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals.

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A337787 Number of addition triangles whose sum is n (version 2).

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A185779 Third accumulation array of Pascal's triangle (as a rectangle), by antidiagonals.

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A337785 Number of addition triangles whose sum is n (version 1).

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A124326 T(n,m) = A007318(n,m) - A077028(n,m).

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