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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 (list; table; graph; refs; listen; history; text; internal format)
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).

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

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(A68625(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

Adjacent sequences:  A168622 A168623 A168624 * A168626 A168627 A168628

KEYWORD

nonn,easy,tabl

AUTHOR

Roger L. Bagula, Dec 01 2009

EXTENSIONS

Definition simplified by R. J. Mathar, Sep 02 2011

STATUS

approved

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Last modified September 16 12:20 EDT 2021. Contains 347472 sequences. (Running on oeis4.)