A103451 -id:A103451 - cp's OEIS frontend

A128715 A131830 + A103451 - A000012 as infinite lower triangular matrices.

1, 2, 2, 3, 1, 3, 4, 2, 2, 4, 5, 3, 5, 3, 5, 6, 4, 9, 9, 4, 6, 7, 5, 14, 19, 14, 5, 7, 8, 6, 20, 34, 34, 20, 6, 8, 9, 7, 27, 55, 69, 55, 27, 7, 9, 10, 8, 35, 83, 125, 125, 83, 35, 8, 10

Offset: 0

Gary W. Adamson, Aug 09 2007

Row sums are (apart from the first row) given by A005126.

			First few rows of the triangle are:
1;
2, 2;
3, 1, 3;
4, 2, 2, 4;
5, 3, 5, 3, 5;
6, 4, 9, 9, 4, 6;
7, 5, 14, 19, 14, 5, 7;
8, 6, 20, 34, 34, 20, 6, 8;
9, 7, 27, 55, 69, 55, 27, 7, 9;
...

Cf. A131830, A103451, A000012, A005126.

A132048 3*A007318 - A103451 - A000012.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 8, 8, 1, 1, 11, 17, 11, 1, 1, 14, 29, 29, 14, 1, 1, 17, 44, 59, 44, 17, 1, 1, 20, 62, 104, 104, 62, 20, 1, 1, 23, 83, 167, 209, 167, 83, 23, 1, 1, 26, 107, 251, 377, 377, 251, 107, 26, 1

Offset: 0

Gary W. Adamson, Aug 08 2007

Row sums = A077802: (1, 2, 7, 18, 41, 88, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  5,  1;
  1,  8,  8,  1;
  1, 11, 17, 11,  1;
  1, 14, 29, 29, 14,  1;
  1, 17, 44, 59, 44, 17,  1;
  ...

Cf. A007318, A103451, A000012, A077802.

3*A007318 - A103451 - A000012 as infinite lower triangular matrices.

A132071 A007318 + A002024 - A103451 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 7, 7, 4, 5, 9, 11, 9, 5, 6, 11, 16, 16, 11, 6, 7, 13, 22, 27, 22, 13, 7, 8, 15, 29, 43, 43, 29, 15, 8, 9, 17, 37, 65, 79, 65, 37, 17, 9, 10, 19, 46, 94, 136, 136, 94, 46, 19, 10

Offset: 0

Gary W. Adamson, Aug 09 2007

Row sums = A132072: (1, 4, 11, 22, 39, 66, 111, ...).

			First few rows of the triangle:
  1;
  2,  2;
  3,  5,  3;
  4,  7,  7,  4;
  5,  9, 11,  9,  5;
  6, 11, 16, 16, 11,  6;
  7, 13, 22, 27, 22, 13,  7;
  ...

Cf. A007318, A002024, A103451, A132072.

A132073 A007318 + A131821 - A103451 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 7, 5, 5, 6, 6, 11, 11, 6, 6, 7, 7, 16, 21, 16, 7, 7, 8, 8, 22, 36, 36, 22, 8, 8, 9, 9, 29, 57, 71, 57, 29, 9, 9, 10, 10, 37, 85, 127, 127, 85, 37, 10, 10

Offset: 0

Gary W. Adamson, Aug 09 2007

Row sums = A132074: (1, 4, 9, 16, 27, 46, 81, 148, 279, ...).

			First few rows of the triangle:
   1;
   2,  2;
   3,  3,  3;
   4,  4,  4,  4;
   5,  5,  7,  5,   5;
   6,  6, 11, 11,   6,   6;
   7,  7, 16, 21,  16,   7,  7;
   8,  8, 22, 36,  36,  22,  8,  8;
   9,  9, 29, 57,  71,  57, 29,  9,  9;
  10, 10, 37, 85, 127, 127, 85, 37, 10, 10;
  ...

Cf. A007318, A131821, A103451, A132074.

A134870 A051731 + A000012 - A103451.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 14 2007

Row sums = A134871: (1, 2, 3, 5, 5, 8, 7, 10, 10, 12, 11, ...).

			First few rows of the triangle:
  1;
  1, 1;
  1, 1, 1;
  1, 2, 1, 1;
  1, 1, 1, 1, 1;
  1, 2, 2, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1;
  1, 2, 1, 2, 1, 1, 1, 1;
  ...

Cf. A051731, A134871, A103451, A000012.

A051731 + A000012 - A103451 as infinite lower triangular matrices.

A144515 Triangle read by rows: A051731 * A103451.

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 3, 1, 0, 1, 2, 0, 0, 0, 1, 4, 1, 1, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 4, 1, 0, 1, 0, 0, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 1, 4, 1, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 0

Gary W. Adamson, Nov 21 2007

Row sums = A114003: (1, 3, 3, 5, 3, 7, 3, 7, 5, 7,...).

Left border = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4,...).

			First few rows of the triangle are:
1;
2, 1;
2, 0, 1;
3, 1, 0, 1;
2, 0, 0, 0, 1;
4, 1, 1, 0, 0, 1;
2, 0, 0, 0, 0, 0, 1;
...

Cf. A114003, A000005, A103451.

A051731 * A103451 as infinite lower triangular matrices. Left border of A051731 (all 1's) is replaced with A000005: (1, 2, 2, 3, 2, 4,...).

A132808 A001263 * A103451 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 6, 6, 1, 42, 10, 20, 10, 1, 132, 15, 50, 50, 15, 1, 429, 21, 105, 175, 105, 21, 1, 1430, 28, 196, 490, 490, 196, 28, 1, 4862, 36, 336, 1176, 1764, 1176, 336, 36, 1, 16796, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 58786, 55, 825, 4950, 13860, 19404, 13860, 4950, 825, 55, 1

Offset: 1

Gary W. Adamson, Aug 31 2007

Replace left border of the Narayana triangle A001263 with the Catalan sequence A000108 starting (1, 2, 5, 14, 42, ...).

Row sums = A131428 starting (1, 3, 9, 27, 83, ...).

			First few rows of the triangle:
    1;
    2,  1
    5,  3,  1;
   14,  6,  6,  1;
   42, 10, 20, 10,  1;
  132, 15, 50, 50, 15, 1;
  ...

Cf. A000108, A001263, A103451.

Definition corrected by Philippe Deléham, Oct 11 2007

a(40) split and more terms from Georg Fischer, May 29 2023

A097806 Riordan array (1+x, 1) read by rows.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Paul Barry, Aug 25 2004

Pair sum operator. Columns have g.f. (1+x)*x^k. Row sums are A040000. Diagonal sums are (1,1,1,....). Riordan inverse is (1/(1+x), 1). A097806 = B*A059260^(-1), where B is the binomial matrix.
Triangle T(n,k), 0<=k<=n, read by rows given by [1, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, May 01 2007
Table T(n,k) read by antidiagonals. T(n,1) = 1, T(n,2) = 1, T(n,k) = 0, k > 2. - Boris Putievskiy, Jan 17 2013

			Rows begin {1}, {1,1}, {0,1,1}, {0,0,1,1}...
From _Boris Putievskiy_, Jan 17 2013: (Start)
The start of the sequence as table:
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
1..1..0..0..0..0..0...
. . .
The start of the sequence as triangle array read by rows:
  1;
  1, 1;
  0, 1, 1;
  0, 0, 1, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 1, 1; . . .
Row number r (r>4) contains (r-2) times '0' and 2 times '1'. (End)

Michael De Vlieger, Table of n, a(n) for n = 0..10010 (Rows 0 <= n <= 140)
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

[k eq n or k eq n-1 select 1 else 0: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 11 2019

A097806 := proc(n,k)
    if k =n or k=n-1 then
        1;
    else
        0;
    end if;
end proc: # R. J. Mathar, Jun 20 2015

Table[Boole[n <= # <= n+1] & /@ Range[n+1], {n, 0, 15}] // Flatten (* or *)
Table[Floor[(# +2)/(n+2)] & /@ Range[n+1], {n, 0, 15}] // Flatten (* Michael De Vlieger, Jul 21 2016 *)

T(n, k) = if(k==n || k==n-1, 1, 0); \\ G. C. Greubel, Jul 11 2019

def T(n, k):
    if (k==n or k==n-1): return 1
    else: return 0
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Jul 11 2019

T(n, k) = if(n=k or n-k=1, 1, 0).
a(n) = A103451(n+1). - Philippe Deléham, Oct 16 2007
From Boris Putievskiy, Jan 17 2013: (Start)
a(n) = floor((A002260(n)+2)/(A003056(n)+2)), n > 0.
a(n) = floor((i+2)/(t+2)), n > 0,
where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)
G.f.: (1+x)/(1-x*y). - R. J. Mathar, Aug 11 2015

A132044 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, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 1, 1, 4, 9, 9, 4, 1, 1, 5, 14, 19, 14, 5, 1, 1, 6, 20, 34, 34, 20, 6, 1, 1, 7, 27, 55, 69, 55, 27, 7, 1, 1, 8, 35, 83, 125, 125, 83, 35, 8, 1, 1, 9, 44, 119, 209, 251, 209, 119, 44, 9, 1

Offset: 0

Gary W. Adamson, Aug 08 2007

Row sums = A132045: (1, 2, 3, 6, 13, 28, 59, ...).

The triangle sequences having the form T(n,k,q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 have the row sums Sum_{k=0..n} T(n,k,q) = 2^(n-2)*q^n + 2^n - (n-1) - (5/4)*[n=0] -(q/2)*[n=1]. - G. C. Greubel, Feb 12 2021

			First few rows of the triangle:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 3,  5,  3,  1;
  1, 4,  9,  9,  4,  1;
  1, 5, 14, 19, 14,  5,  1;
  1, 6, 20, 34, 34, 20,  6, 1;
  1, 7, 27, 55, 69, 55, 27, 7, 1;

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

Cf. A007318, A103451, A132045.

Cf. this sequence (q=0), A173075 (q=1), A173046 (q=2), A173047 (q=3).

T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) + q^n*Binomial(n-2,k-1) -1 >;
[T(n,k,0): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 12 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 (* Roger L. Bagula, Feb 08 2010 *)

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

T(n, k) = A007318(n,k) + A103451(n,k) - A000012(n,k), an infinite lower triangular matrix.
T(n, k) = binomial(n, k) - 1, with T(n,0) = T(n,n) = 1. - Roger L. Bagula, Feb 08 2010
From G. C. Greubel, Feb 12 2021: (Start)
T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 0.
Sum_{k=0..n} T(n, k, 0) = 2^n - (n-1) - [n=0]. (End)

A132046 Triangle read by rows: T(n,0) = T(n,n) = 1, and T(n,k) = 2*binomial(n,k) for 1 <= k <= n - 1.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 12, 8, 1, 1, 10, 20, 20, 10, 1, 1, 12, 30, 40, 30, 12, 1, 1, 14, 42, 70, 70, 42, 14, 1, 1, 16, 56, 112, 140, 112, 56, 16, 1, 1, 18, 72, 168, 252, 252, 168, 72, 18, 1, 1, 20, 90, 240, 420, 504, 420, 240, 90, 20, 1

Offset: 0

Gary W. Adamson, Aug 08 2007

T(2*n,n) is A100320 (with Hankel transform A144704). - Paul Barry, Sep 19 2008
Double the internal elements of Pascal's triangle. - Paul Barry, Jan 07 2009
Coefficients of 2*(x + 1)^n - (x^n + 1) as a triangle (except for the very first term). - Thomas Baruchel, Jun 02 2018

			First few rows of the triangle are:
  1;
  1,  1;
  1,  4,  1;
  1,  6,  6,  1;
  1,  8, 12,  8,  1;
  1, 10, 20, 20, 10,  1;
  1, 12, 30, 40, 30, 12,  1;
  1, 14, 42, 70, 70, 42, 14, 1;
  ...

Row sums: A095121.
Cf. A007318, A103451, A132046.
Cf. A154327 (diagonal sums). [Paul Barry, Jan 07 2009]
Cf. A141540.

T[n_, k_] := If[n == k || k == 0, 1, If[k <= n, 2 Binomial[n, k], 0]]
Flatten[Table[T[n, k], {n, 0, 20}, {k, 0, n}]] (* Emanuele Munarini, May 15 2018 *)

T(n, k) := if k = 0 or k = n then 1 else 2*binomial(n, k)$
create_list(T(n, k), n, 0, 20, k, 0, n); /* Franck Maminirina Ramaharo, Jan 03 2019 */

T(n,k) = 2*A007318(n,k) - A103451(n,k).
T(n,k) = [k<=n] (0^(n + k) + C(n,k)*(2 - 0^(n - k) - 0^k)). - Paul Barry, Sep 19 2008
T(n,k) = A007318(n,k)*A154325(n,k). - Paul Barry, Jan 07 2009
From Emanuele Munarini, May 15 2018: (Start)
G.f.: (1 - t - x*t + 3*x*t^2 - x*t^3 - x^2*t^3)/((1 - t)*(1 - x*t)*(1 - t - x*t)).
T(n+3,k+2) = 2*T(n+2,k+2) - T(n+1,k+2) + 2*T(n+2,k+1) - 3*T(n+1,k+1) - T(n+1,k) + T(n,k+1) + T(n,k), except for n = 0 and k = 0. (End)
E.g.f.: 1 - exp(t) - exp(t*x) + 2*exp(t*(1 + x)). - Franck Maminirina Ramaharo, Jan 02 2019

cp's OEIS Frontend

A128715 A131830 + A103451 - A000012 as infinite lower triangular matrices.

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A132048 3*A007318 - A103451 - A000012.

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A132071 A007318 + A002024 - A103451 as infinite lower triangular matrices.

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A132073 A007318 + A131821 - A103451 as infinite lower triangular matrices.

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A134870 A051731 + A000012 - A103451.

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A144515 Triangle read by rows: A051731 * A103451.

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A132808 A001263 * A103451 as infinite lower triangular matrices.

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A097806 Riordan array (1+x, 1) read by rows.

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A132044 Triangle T(n,k) = binomial(n, k) - 1 with T(n,0) = T(n,n) = 1, read by rows.

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