A173076 -id:A173076 - cp's OEIS frontend

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.

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)

A173077 Triangle T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 12, 23, 12, 1, 1, 13, 36, 36, 13, 1, 1, 32, 122, 181, 122, 32, 1, 1, 33, 155, 304, 304, 155, 33, 1, 1, 88, 513, 1270, 1689, 1270, 513, 88, 1, 1, 89, 602, 1784, 2960, 2960, 1784, 602, 89, 1, 1, 252, 1988, 6923, 13817, 17261, 13817, 6923, 1988, 252, 1

Offset: 0

Roger L. Bagula, Feb 09 2010

			Triangle starts:
  1;
  1,   1;
  1,   4,    1;
  1,   5,    5,    1;
  1,  12,   23,   12,     1;
  1,  13,   36,   36,    13,     1;
  1,  32,  122,  181,   122,    32,     1;
  1,  33,  155,  304,   304,   155,    33,    1;
  1,  88,  513, 1270,  1689,  1270,   513,   88,    1;
  1,  89,  602, 1784,  2960,  2960,  1784,  602,   89,   1;
  1, 252, 1988, 6923, 13817, 17261, 13817, 6923, 1988, 252, 1;
  ...
Row sums: 1, 2, 6, 12, 49, 100, 491, 986, 5433, 10872, 63223, ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A132044 (q=0), A173075 (q=1), A173076 (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^(Floor(n/2))*Binomial(n-2,k-1) -1 >;
[T(n,k,3): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021

T[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + 3^Floor[n/2] Binomial[n-2, k- 1]];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

def T(n,k,q): return 1 if (k==0 or k==n) else binomial(n,k) + q^(n//2)*binomial(n-2,k-1) -1
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 09 2021

T(n, k, q) = binomial(n, k) - 1 + q^floor(n/2)*binomial(n-2, k-1) with T(n, 0, q) = T(n, n, q) = 1 and q = 3.

cp's OEIS Frontend

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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