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A168625
Triangle T(n,k) = 8*binomial(n,k) - 7 with columns 0 <= k <= n.
4
1, 1, 1, 1, 9, 1, 1, 17, 17, 1, 1, 25, 41, 25, 1, 1, 33, 73, 73, 33, 1, 1, 41, 113, 153, 113, 41, 1, 1, 49, 161, 273, 273, 161, 49, 1, 1, 57, 217, 441, 553, 441, 217, 57, 1, 1, 65, 281, 665, 1001, 1001, 665, 281, 65, 1, 1, 73, 353, 953, 1673, 2009, 1673, 953, 353, 73, 1
OFFSET
0,5
COMMENTS
Triangle T(n,k): the coefficient [x^k] of the polynomial 8*(x+1)^n -7*( x^(n+1) - 1)/(x-1).
FORMULA
T(n,k) = [x^k] ( 8*(x+1)^n-7*Sum_{s=0..n} x^s ) = 8*A007318(n,k) - 7. - R. J. Mathar, Sep 02 2011
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 9, 1;
1, 17, 17, 1;
1, 25, 41, 25, 1;
1, 33, 73, 73, 33, 1;
1, 41, 113, 153, 113, 41, 1;
1, 49, 161, 273, 273, 161, 49, 1;
1, 57, 217, 441, 553, 441, 217, 57, 1;
1, 65, 281, 665, 1001, 1001, 665, 281, 65, 1;
1, 73, 353, 953, 1673, 2009, 1673, 953, 353, 73, 1;
MAPLE
A168625:= (n, k) -> 8*binomial(n, k) -7; seq(seq(A168625(n, k), k = 0..n), n = 0.. 10); # G. C. Greubel, Mar 12 2020
MATHEMATICA
m = 8; p[x_, n_]:= FullSimplify[ExpandAll[m*(x+1)^n -(m-1)(x^(n+1) -1)/(x-1)]];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten
Table[8*Binomial[n, k] -7, {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)
PROG
(Magma) [8*Binomial(n, k) -7: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020
(Sage) [[8*binomial(n, k) -7 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020
CROSSREFS
Sequence m*binomial(n,k) - (m-1): A007318 (m=1), A109128 (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), A131067 (m=7), this sequence (m=8).
Sequence in context: A092578 A331247 A128060 * A143681 A081582 A174346
KEYWORD
nonn,easy,tabl
AUTHOR
Roger L. Bagula, Dec 01 2009
EXTENSIONS
Definition simplified by R. J. Mathar, Sep 02 2011
STATUS
approved