A132823 -id:A132823 - cp's OEIS frontend

A132824 Row sums of triangle A132823.

1, 2, 2, 4, 10, 24, 54, 116, 242, 496, 1006, 2028, 4074, 8168, 16358, 32740, 65506, 131040, 262110, 524252, 1048538, 2097112, 4194262, 8388564, 16777170, 33554384, 67108814, 134217676, 268435402, 536870856, 1073741766, 2147483588, 4294967234, 8589934528

Offset: 0

Gary W. Adamson, Sep 02 2007

			a(4) = 10 = sum of row 4 terms of triangle A132823: (1 + 2 + 4 + 2 + 1).
a(3) = 4 = (1, 3, 3, 1) dot (1, 1, -1, 3) = (1 + 3 -3 + 3).

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000325, A132823, A183155.

Essentially the same as A132732.

A132824:=n->`if`(n=0, 1, 2+2^n-2*n); seq(A132824(n), n=0..30); # Wesley Ivan Hurt, Jun 06 2014

a[0] = 1; a[n_] := 2 + 2^n - 2*n; Table[a[n], {n, 0, 30}] (* Wesley Ivan Hurt, Jun 06 2014 *)

Binomial transform of [1, 1, -1, 3, -1, 3, -1, 3, -1, 3, ...].
For n > 0, a(n) = 2 + 2^n - 2*n = 1 + A183155(n-1). - R. J. Mathar, Apr 04 2012
From Colin Barker, Jun 06 2014: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n > 3.
G.f.: -(4*x^3-x^2-2*x+1)/((x-1)^2*(2*x-1)). (End)
For n > 1, a(n) = A132732(n-1). - Jeppe Stig Nielsen, Dec 29 2017
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(exp(x) - 2*(x - 1)) - 2.
a(n) = 2*A000325(n-1) for n >= 1. (End)

A323846 Array read by antidiagonals: T(m,n) = number of m X n matrices M with entries {0,1,2} that have M_{1,1}=0, M_{m,n}=2, are such that the rows and columns are monotonic without jumps of 2, and satisfy M_{(i+1),(j+1)} = M_{i,j} + (0 or 1).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 3, 4, 4, 3, 6, 16, 25, 16, 6, 10, 41, 94, 94, 41, 10, 15, 85, 266, 386, 266, 85, 15, 21, 155, 632, 1247, 1247, 632, 155, 21, 28, 259, 1332, 3423, 4657, 3423, 1332, 259, 28, 36, 406, 2570, 8342, 14795, 14795, 8342, 2570, 406, 36, 45, 606, 4631, 18546, 41586, 54219, 41586, 18546, 4631, 606, 45

Offset: 1

N. J. A. Sloane, Feb 06 2019

The monotonicity condition requires that M_{(i+1),j} = M_{i,j} + (0 or 1); M_{i,(j+1)} = M_{i,j} + (0 or 1).
These matrices can be cut into three connected pieces, containing the 0's, 1's, and 2's; there are two vertex-disjoint paths from the north-and-east edges of the matrix to the south-and-west edges.
Row (or column) n >= 1 has a linear recurrence (with constant coefficients) of order 2n+1. - Alois P. Heinz, Feb 07 2019

			Array begins:
    0   0    1    3     6    10 ...
    0   0    4   16    41    85 ...
    1   4   25   94   266   632 ...
    3  16   94  386  1247  3423 ...
    6  41  266 1247  4657 14795 ...
   10  85  632 3427 14795 54219 ...
...
The 4 examples when m=2 and n=3 are
    011   011  012   012
    012   112  012   112

D. E. Knuth, Email to N. J. A. Sloane, Feb 05 2019.

Alois P. Heinz, Antidiagonals n = 1..80, flattened

Rows 1-10 give: A000217(n-2), A323847, A323967, A323968, A323969, A323970, A323971, A323972, A323973, A323974.
Main diagonal gives A306322.
Cf. A132823, A252876, A229428.

More terms from Alois P. Heinz, Feb 07 2019

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

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

cp's OEIS Frontend

A132824 Row sums of triangle A132823.

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A323846 Array read by antidiagonals: T(m,n) = number of m X n matrices M with entries {0,1,2} that have M_{1,1}=0, M_{m,n}=2, are such that the rows and columns are monotonic without jumps of 2, and satisfy M_{(i+1),(j+1)} = M_{i,j} + (0 or 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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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.

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