|
| |
|
|
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.
|
|
4
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
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
|
|
|
LINKS
|
G. C. Greubel, Rows n = 0..100 of the triangle, flattened
|
|
|
FORMULA
|
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
|
|
|
EXAMPLE
|
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;
|
|
|
MATHEMATICA
|
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 *)
|
|
|
PROG
|
(Sage)
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
(Magma)
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
|
|
|
CROSSREFS
|
Cf. A132044 (q=0), A173075 (q=1), A173046 (q=2), this sequence (q=3).
Sequence in context: A146765 A190152 A154984 * A173045 A176491 A008958
Adjacent sequences: A173044 A173045 A173046 * A173048 A173049 A173050
|
|
|
KEYWORD
|
nonn,tabl
|
|
|
AUTHOR
|
Roger L. Bagula, Feb 08 2010
|
|
|
EXTENSIONS
|
Edited by G. C. Greubel, Feb 16 2021
|
|
|
STATUS
|
approved
|
| |
|
|
