A132044 -id:A132044 - cp's OEIS frontend

A132045 Row sums of triangle A132044.

1, 2, 3, 6, 13, 28, 59, 122, 249, 504, 1015, 2038, 4085, 8180, 16371, 32754, 65521, 131056, 262127, 524270, 1048557, 2097132, 4194283, 8388586, 16777193, 33554408, 67108839, 134217702, 268435429, 536870884, 1073741795, 2147483618, 4294967265, 8589934560

Offset: 0

Gary W. Adamson, Aug 08 2007

Apart from initial terms, and with a change of offset, same as A095768. - Jon E. Schoenfield, Jun 15 2017

			a(4) = 13 = sum of row 4 terms of triangle A132044: (1 + 3 + 5 + 3 + 1).
a(4) = 13 = (1, 4, 6, 4, 1) dot (1, 1, 0, 2, 0) = (1 + 4 + 0 + 8 + 0).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000325, A095768, A132044.

[1] cat [2^n -n +1: n in [1..35]]; // G. C. Greubel, Feb 12 2021

Table[2^n -(n-1) -Boole[n==0], {n, 0, 35}] (* G. C. Greubel, Feb 12 2021 *)

Vec((1-2*x+2*x^3)/((1-x)^2*(1-2*x)) + O(x^100)) \\ Colin Barker, Mar 14 2014

[1]+[2^n -n +1 for n in (1..35)] # G. C. Greubel, Feb 12 2021

Binomial transform of (1, 1, 0, 2, 0, 2, 0, 2, 0, 2, ...).
For n>=1, a(n) = 2^n - n + 1 = A000325(n) + 1. - Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 17 2009. (Corrected by Franklin T. Adams-Watters, Jan 17 2009)
E.g.f.: U(0) - 1, where U(k) = 1 - x/(2^k + 2^k/(x - 1 - x^2*2^(k+1)/(x*2^(k+1) + (k+1)/U(k+1)))). - Sergei N. Gladkovskii, Dec 01 2012
From Colin Barker, Mar 14 2014: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n>3.
G.f.: (1-2*x+2*x^3) / ((1-x)^2*(1-2*x)). (End)

A173075 T(n,k) = binomial(n, k) - 1 + q^(floor(n/2))*binomial(n-2, k-1) for 0 < k < n with T(n,0) = T(n,n) = 1 and q = 1. Triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 12, 12, 5, 1, 1, 6, 18, 25, 18, 6, 1, 1, 7, 25, 44, 44, 25, 7, 1, 1, 8, 33, 70, 89, 70, 33, 8, 1, 1, 9, 42, 104, 160, 160, 104, 42, 9, 1, 1, 10, 52, 147, 265, 321, 265, 147, 52, 10, 1

Offset: 0

Roger L. Bagula, Feb 09 2010

Rows two through six appear in the table on p. 8 of Getzler. Cf. also A167763. - Tom Copeland, Jan 22 2020

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

			Triangle begins:
  1,
  1,  1;
  1,  2,  1;
  1,  3,  3,   1;
  1,  4,  7,   4,   1;
  1,  5, 12,  12,   5,   1;
  1,  6, 18,  25,  18,   6,   1;
  1,  7, 25,  44,  44,  25,   7,   1;
  1,  8, 33,  70,  89,  70,  33,   8,  1;
  1,  9, 42, 104, 160, 160, 104,  42,  9,  1;
  1, 10, 52, 147, 265, 321, 265, 147, 52, 10, 1;
  ...
Row sums: {1, 2, 4, 8, 17, 36, 75, 154, 313, 632, 1271, ...}.

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
E. Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996.

Cf. A132044 (q=0), this sequence (q=1), A173076 (q=2), A173077 (q=3).
Cf. A132044 (p=0), this sequence (p=1), A173046 (p=2), A173047 (p=3).
Cf. A167763.

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

T[n_, m_]:= If[m==0 || m==n, 1, Binomial[n, m] - 1 + Binomial[n-2, m-1]];
Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten

T(n,k)={if(k<=0||k>=n, k==0||k==n, binomial(n,k) - 1 + binomial(n-2, k-1))} \\ Andrew Howroyd, Jan 22 2020

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

T(n, k) = binomial(n, k) - 1 + binomial(n-2, k-1) for 0 < k < n.
T(n, 0) = T(n, n) = 1.
From G. C. Greubel, Feb 12 2021: (Start)
T(n, k, p) = binomial(n, k) + p^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and p = 1.
Sum_{k=0..n} T(n, k, 1) = 2^(n-2) + 2^n - (n-1) - (5/4)*[n=0] -(1/2)*[n=1]. (End)

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

A173046 Triangle 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 = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 10, 10, 1, 1, 19, 37, 19, 1, 1, 36, 105, 105, 36, 1, 1, 69, 270, 403, 270, 69, 1, 1, 134, 660, 1314, 1314, 660, 134, 1, 1, 263, 1563, 3895, 5189, 3895, 1563, 263, 1, 1, 520, 3619, 10835, 18045, 18045, 10835, 3619, 520, 1, 1, 1033, 8236, 28791, 57553, 71931, 57553, 28791, 8236, 1033, 1

Offset: 0

Roger L. Bagula, Feb 08 2010

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 16 2021

			Triangle begins as:
  1;
  1,    1;
  1,    5,    1;
  1,   10,   10,     1;
  1,   19,   37,    19,     1;
  1,   36,  105,   105,    36,     1;
  1,   69,  270,   403,   270,    69,     1;
  1,  134,  660,  1314,  1314,   660,   134,     1;
  1,  263, 1563,  3895,  5189,  3895,  1563,   263,    1;
  1,  520, 3619, 10835, 18045, 18045, 10835,  3619,  520,    1;
  1, 1033, 8236, 28791, 57553, 71931, 57553, 28791, 8236, 1033, 1;

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

Cf. A132044 (q=0), A173075 (q=1), this sequence (q=2), A173047 (q=3).

Cf. A000302, A132045.

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,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

T[n_, m_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] +(q^n)*Binomial[n-2, k-1] -1];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 16 2021 *)

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,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 16 2021

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 = 2.

Sum_{k=0..n} T(n, k, 2) = 4^(n-1) + 2^n - (n-1) - (5/4)*[n=0] = A000302(n-1) + A132045(n) - (5/4)*[n=0]. - [n=1]. - G. C. Greubel, Feb 16 2021

Edited by G. C. Greubel, Feb 16 2021

A173047 Triangle 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 = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 29, 29, 1, 1, 84, 167, 84, 1, 1, 247, 738, 738, 247, 1, 1, 734, 2930, 4393, 2930, 734, 1, 1, 2193, 10955, 21904, 21904, 10955, 2193, 1, 1, 6568, 39393, 98470, 131289, 98470, 39393, 6568, 1, 1, 19691, 137816, 413426, 689030, 689030, 413426, 137816, 19691, 1

Offset: 0

Roger L. Bagula, Feb 08 2010

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 16 2021

			Ttiangle begins as:
  1;
  1,     1;
  1,    10,      1;
  1,    29,     29,      1;
  1,    84,    167,     84,      1;
  1,   247,    738,    738,    247,      1;
  1,   734,   2930,   4393,   2930,    734,      1;
  1,  2193,  10955,  21904,  21904,  10955,   2193,      1;
  1,  6568,  39393,  98470, 131289,  98470,  39393,   6568,     1;
  1, 19691, 137816, 413426, 689030, 689030, 413426, 137816, 19691, 1;

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

Cf. A132044 (q=0), A173075 (q=1), A173046 (q=2), this sequence (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,3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] +(q^n)*Binomial[n-2, k-1] -1];
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 16 2021 *)

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,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 16 2021

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 = 3.

Sum_{k=0..n} T(n, k, 3) = (1/4)*(6^n + 2^(n+2) - 4*(n-1) - 5*[n=0] - 6*[n=1]). - G. C. Greubel, Feb 16 2021

Edited 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

A132729 Triangle T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 9, 5, 1, 1, 7, 17, 17, 7, 1, 1, 9, 27, 37, 27, 9, 1, 1, 11, 39, 67, 67, 39, 11, 1, 1, 13, 53, 109, 137, 109, 53, 13, 1, 1, 15, 69, 165, 249, 249, 165, 69, 15, 1, 1, 17, 87, 237, 417, 501, 417, 237, 87, 17, 1, 1, 19, 107, 327, 657, 921, 921, 657, 327, 107, 19, 1

Offset: 0

Gary W. Adamson, Aug 26 2007

			First few rows of the triangle are:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  3,  1;
  1,  5,  9,  5,  1;
  1,  7, 17, 17,  7,  1;
  1,  9, 27, 37, 26,  9,  1;
  1, 11, 39, 67, 67, 39, 11, 1;

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

Cf. A007318, A132044, A132730.

T:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) - 3 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021

T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n, k] - 3];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 13 2021 *)

def T(n,k): return 1 if (k==0 or k==n) else 2*binomial(n,k) - 3
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

T(n, k) = 2*A132044(n, k) - 1.
From G. C. Greubel, Feb 13 2021: (Start)
T(n, k) = 2*binomial(n, k) - 3 with T(n, 0) = T(n, n) = 1.
Sum_{k=0..n} T(n, k) = 2^(n+1) - 3*n + 1 - 2*[n=0] = A132730(n). (End)

More terms added by G. C. Greubel, Feb 13 2021

cp's OEIS Frontend

A132045 Row sums of triangle A132044.

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A173075 T(n,k) = binomial(n, k) - 1 + q^(floor(n/2))*binomial(n-2, k-1) for 0 < k < n with T(n,0) = T(n,n) = 1 and q = 1. Triangle read by rows.

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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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A173046 Triangle 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 = 2, read by rows.

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