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A132737
Triangle T(n,k) = 2*binomial(n,k) + 1, read by rows.
2
1, 1, 1, 1, 5, 1, 1, 7, 7, 1, 1, 9, 13, 9, 1, 1, 11, 21, 21, 11, 1, 1, 13, 31, 41, 31, 13, 1, 1, 15, 43, 71, 71, 43, 15, 1, 1, 17, 57, 113, 141, 113, 57, 17, 1, 1, 19, 73, 169, 253, 253, 169, 73, 19, 1, 1, 21, 91, 241, 421, 505, 421, 241, 91, 21, 1, 1, 23, 111, 331, 661, 925, 925, 661, 331, 111, 23, 1
OFFSET
0,5
FORMULA
T(n, k) = 2*A132735(n, k) - 1, an infinite lower triangular matrix.
T(n,0) = T(n,n) = 1; otherwise T(n,k) = 2*C(n,k) + 1. - Franklin T. Adams-Watters, Jul 06 2009
Sum_{k=0..n} T(n, k) = 2^(n+1) + n - 3 + 2*[n=0] = A132738(n). - G. C. Greubel, Feb 15 2021
EXAMPLE
First few rows of the triangle are:
1;
1, 1;
1, 5, 1;
1, 7, 7, 1;
1, 9, 13, 9, 1;
1, 11, 21, 21, 11, 1;
1, 13, 31, 41, 31, 13, 1;
1, 15, 43, 71, 71, 43, 15, 1;
...
MATHEMATICA
T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n, k] +1];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 15 2021 *)
PROG
(Sage)
def A132737(n, k): return 1 if (k==0 or k==n) else 2*binomial(n, k) + 1
flatten([[A132737(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 15 2021
(Magma)
A132737:= func< n, k | k eq 0 or k eq n select 1 else 2*Binomial(n, k) +1 >;
[A132737(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 15 2021
CROSSREFS
Sequences of the form 2*binomial(n,k) + q: A132729 (q=-3), A132731 (q=-2), A109128 (q=-1), A132046 (q=0), this sequence (q=1).
Sequence in context: A111720 A029646 A152721 * A137754 A131404 A297943
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Aug 26 2007
EXTENSIONS
Extended by Franklin T. Adams-Watters, Jul 06 2009
STATUS
approved