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A103247
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Triangle read by rows: T(n,k) is the coefficient of x^k (0<=k<=n) in the monic characteristic polynomial of the n X n matrix with 3's on the diagonal and 1's elsewhere (n>=1). Row 0 consists of the single term 1.
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5
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1, -3, 1, 8, -6, 1, -20, 24, -9, 1, 48, -80, 48, -12, 1, -112, 240, -200, 80, -15, 1, 256, -672, 720, -400, 120, -18, 1, -576, 1792, -2352, 1680, -700, 168, -21, 1, 1280, -4608, 7168, -6272, 3360, -1120, 224, -24, 1, -2816, 11520, -20736, 21504, -14112, 6048, -1680, 288, -27, 1, 6144, -28160, 57600, -69120
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Row sums of the unsigned triangle yield A006234. The unsigned triangle is the mirror image of A103407.
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FORMULA
| Appears to be the matrix product (I-S)*P^(-2), where I is the identity, P is Pascal's triangle A007318 and S is A132440, the infinitesimal generator of P. Cf. A055137 (= (I-S)*P) and A103283 (= (I-S)*P^(-1)) - Peter Bala, Nov 28 2011
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EXAMPLE
| The monic characteristic polynomial of the matrix [3 1 1 / 1 3 1 / 1 1 3] is x^3 - 9x^2 + 24x - 20; so T(3,0)=-20, T(3,1)=24, T(3,2)=-9, T(3,3)=1.
Triangle begins:
1;
-3,1;
8,-6,1;
-20,24,-9,1;
48,-80,48,-12,1;
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MAPLE
| with(linalg): a:=proc(i, j) if i=j then 3 else 1 fi end: 1; for n from 1 to 10 do seq(coeff(expand(x*charpoly(matrix(n, n, a), x)), x^k), k=1..n+1) od; # yields the sequence in triangular form
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CROSSREFS
| Cf. A006234, A103407.
Sequence in context: A206800 A188939 A062196 * A030523 A123965 A124025
Adjacent sequences: A103244 A103245 A103246 * A103248 A103249 A103250
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KEYWORD
| sign,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 19 2005
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