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A130595
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Triangle read by rows: lower triangular matrix which is inverse to Pascal's triangle (A007318) regarded as a lower triangular matrix.
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27
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1, -1, 1, 1, -2, 1, -1, 3, -3, 1, 1, -4, 6, -4, 1, -1, 5, -10, 10, -5, 1, 1, -6, 15, -20, 15, -6, 1, -1, 7, -21, 35, -35, 21, -7, 1, 1, -8, 28, -56, 70, -56, 28, -8, 1, -1, 9, -36, 84, -126, 126, -84, 36, -9, 1, 1, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1, -1, 11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Triangle T(n,k), read by rows,given by [-1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 .
Coefficients of the polynomials generated by the e.g.f. exp(x*t)*exp(-t). [From Peter Luschny (peter(AT)luschny.de), Jul 13 2009]
Riordan array (1/(1+x), x/(1+x)). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 29 2009]
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REFERENCES
| He, Tian-Xiao, and Sprugnoli, Renzo; Sequence characterization of Riordan arrays. Discrete Math. 309 (2009), no. 12, 3962-3974. [From N. J. A. Sloane, Nov 26, 2011]
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FORMULA
| T(n,k)=(-1)^(n-k)*binomial(n,k)=(-1)^(n-k)*A007318(n,k).
T(n,k)=T(n-1,k-1)-T(n-1,k) - From DELEHAM Philippe, Oct 10 2011.
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EXAMPLE
| Triangle begins:
1;
-1, 1;
1, -2, 1;
-1, 3, -3, 1;
1, -4, 6, -4, 1;
-1, 5, -10, 10, -5, 1;
1, -6, 15, -20, 15, -6, 1;
-1, 7, -21, 35, -35, 21, -7, 1;
1, -8, 28, -56, 70, -56, 28, -8, 1;
-1, 9, -36, 84, -126, 126, -84, 36, -9, 1 ;...
Contribution from Peter Luschny (peter(AT)luschny.de), Jul 13 2009: (Start)
+ 1
- 1 + 1 x
+ 1 - 2 x + 1 x^2
- 1 + 3 x - 3 x^2 + 1 x^3
+ 1 - 4 x + 6 x^2 - 4 x^3 + 1 x^4 (End)
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MAPLE
| with(combstruct):for n from 0 to 11 do seq((-1)^(n-m)*count(Combination(n), size=m), m = 0 .. n) od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 09 2008
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MATHEMATICA
| nmax = 11; t[n_, k_] := (-1)^(n-k)*Binomial[n, k]; Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}] ] (* From Jean-François Alcover, Dec 01 2011 *)
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PROG
| (Haskell)
a130595 n = a130595_list !! n
a130595_list = concat $ iterate ([-1, 1] *) [1]
instance Num a => Num [a] where
fromInteger k = [fromInteger k]
(p:ps) + (q:qs) = p + q : ps + qs
ps + qs = ps ++ qs
(p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
_ * _ = []
-- Reinhard Zumkeller, Apr 02 2011
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CROSSREFS
| Cf. A007318.
Sequence in context: A118433 A007318 A108086 * A108363 A076831 A197061
Adjacent sequences: A130592 A130593 A130594 * A130596 A130597 A130598
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KEYWORD
| sign,nice,tabl
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 17 2007
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EXTENSIONS
| Edited by N. J. A. Sloane, Nov 27 2011
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