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A112468
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Riordan array (1/(1-x),x/(1+x)).
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13
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1, 1, 1, 1, 0, 1, 1, 1, -1, 1, 1, 0, 2, -2, 1, 1, 1, -2, 4, -3, 1, 1, 0, 3, -6, 7, -4, 1, 1, 1, -3, 9, -13, 11, -5, 1, 1, 0, 4, -12, 22, -24, 16, -6, 1, 1, 1, -4, 16, -34, 46, -40, 22, -7, 1, 1, 0, 5, -20, 50, -80, 86, -62, 29, -8, 1, 1, 1, -5, 25, -70, 130, -166, 148, -91, 37, -9, 1, 1, 0, 6, -30, 95, -200, 296, -314, 239, -128, 46, -10, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,13
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COMMENTS
| Row sums are A040000. Diagonal sums are A112469. Inverse is A112467. Row sums of k-th power are 1,k+1,k+1,k+1,.... Note that C(n,k)=sum{j=0..n-k, C(n-j-1,n-k-j)}.
Equals row reversal of triangle A112555 up to sign, where log(A112555) = A112555 - I. Unsigned row sums equals A052953 (Jacobsthal numbers + 1). Central terms of even-indexed rows are a signed version of A072547. Sums of squared terms in rows yields A112556, which equals the first differences of the unsigned central terms. - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 20 2006
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REFERENCES
| E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
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FORMULA
| Triangle T(n,k) read by rows: T(n,0)=1, T(n,k)=T(n-1,k-1)-T(n-1,k). - Mats Granvik (mats.granvik(AT)abo.fi), Mar 15 2010
Number triangle T(n, k)=sum{j=0..n-k, C(n-j-1, n-k-j)*(-1)^(n-k-j)}
G.f. of matrix power T^m: (1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x). G.f. of matrix log: x*(1-2*x*y+x^2*y)/(1-x*y)^2/(1-x). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 20 2006
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EXAMPLE
| Triangle starts
1;
1,1;
1,0,1;
1,1,-1,1;
1,0,2,-2,1;
1,1,-2,4,-3,1;
1,0,3,-6,7,-4,1;
Matrix log begins:
0;
1,0;
1,0,0;
1,1,-1,0;
1,1,1,-2,0;
1,1,1,1,-3,0; ...
Production matrix begins
1, 1,
0, -1, 1,
0, 0, -1, 1,
0, 0, 0, -1, 1,
0, 0, 0, 0, -1, 1,
0, 0, 0, 0, 0, -1, 1,
0, 0, 0, 0, 0, 0, -1, 1
[Paul Barry, April 8 2011]
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PROG
| (PARI) {T(n, k)=local(m=1, x=X+X*O(X^n), y=Y+Y*O(Y^k)); polcoeff(polcoeff((1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x), n, X), k, Y)} (Hanna)
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CROSSREFS
| Cf. A174294, A174295, A174296, A174297. - Mats Granvik (mats.granvik(AT)abo.fi), Mar 15 2010
Cf. A112465.
Cf. A112555 (reversed rows), A052953, A072547 (central terms), A112556.
Sequence in context: A031282 A085685 A112465 * A086275 A066855 A175685
Adjacent sequences: A112465 A112466 A112467 * A112469 A112470 A112471
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KEYWORD
| easy,sign,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 06 2005
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