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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.
4
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
0,5
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
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 = 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
EXAMPLE
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;
MATHEMATICA
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 *)
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, 2) 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, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021
CROSSREFS
Cf. A132044 (q=0), A173075 (q=1), this sequence (q=2), A173047 (q=3).
Sequence in context: A356113 A188461 A188474 * A173043 A082046 A132787
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 08 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 16 2021
STATUS
approved