A173046 -id:A173046 - cp's OEIS frontend

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

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)

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)

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

			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

cp's OEIS Frontend

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

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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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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.

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