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A129818
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Riordan array (1/(1+x),x/(1+x)^2), inverse array is A039599 .
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10
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1, -1, 1, 1, -3, 1, -1, 6, -5, 1, 1, -10, 15, -7, 1, -1, 15, -35, 28, -9, 1, 1, -21, 70, -84, 45, -11, 1, -1, 28, -126, 210, -165, 66, -13, 1, 1, -36, 210, -462, 495, -286, 91, -15, 1, -1, 45, -330, 924, -1287, 1001, -455, 120, -17, 1, 1, -55, 495, -1716, 3003, -3003, 1820, -680, 153, -19, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| This sequence is the same as A123970. - T. D. Noe, Sep 30 2011
Row sums : A057078 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 11 2007
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REFERENCES
| Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
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FORMULA
| T(n,k)=(-1)^(n-k)*A085478(n,k)= (-1)^(n-k)*binomial(n+k,2*k) .
Sum_{k, 0<=k<=n}T(n,k)*A000531(k)=n^2, with A000531(0)=0 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 11 2007
Sum_{k, 0<=k<=n}T(n,k)*x^k = A033999(n), A057078(n), A057077(n), A057079(n), A005408(n), A001906(n), A001834(n), A030221(n), A002315(n), A033890(n), A057080(n), A057081(n), A054320(n), A097783(n), A077416(n), A126866(n), A028230(n+1) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2009]
O.g.f. for row polynomials (rising powers in y): (1+x)/(1+(2-y)*x+x^2). W. Lang, Dec 15 2010.
O.g.f. column k with leading zeros (Riordan array, see NAME): (1/(1+x))*(x/(1+x)^2)^k, k>=0. W. Lang, Dec 15 2010.
Recurrences from the Z- and A-sequences for Riordan arrays. See the W. Lang link under A006232 for details and references.
T(n,0) = -1*T(n-1,0), n>=1, from the o.g.f. -1 for the Z-sequence (trivial result).
T(n,k) = sum(A(j)*T(n-1,k-1+j,j=0..n-k), n>=k>=1, with A(j):= A115141(j) = [1,-2,-1,-2,-5,-14,...,],j>=0 (o.g.f. 1/c(x)^2 with the A000108 (Catalan) o.g.f. c(x)). W. Lang, Dec 20 2010.
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EXAMPLE
| Triangle begins:
1;
-1, 1;
1, -3, 1;
-1, 6, -5, 1;
1, -10, 15, -7, 1;
-1, 15, -35, 28, -9, 1;
1, -21, 70, -84, 45, -11, 1;
-1, 28, -126, 210, -165, 66, -13, 1;
1, -36, 210, -462, 495, -286, 91, -15, 1;
-1, 45, -330, 924, -1287, 1001, -455, 120, -17, 1;
1, -55, 495, -1716, 3003, -3003, 1820, -680, 153, -19, 1 ;...
Recurrence from the A-sequence A115141:
15 = T(4,2) = 1*6 + (-2)*(-5) + (-1)*1.
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MATHEMATICA
| max = 10; Flatten[ CoefficientList[#, y] & /@ CoefficientList[ Series[ (1 + x)/(1 + (2 - y)*x + x^2), {x, 0, max}], x]] (* From Jean-François Alcover, Sep 29 2011, after W. Lang *)
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CROSSREFS
| Sequence in context: A102036 A121524 A103141 * A085478 A123970 A055898
Adjacent sequences: A129815 A129816 A129817 * A129819 A129820 A129821
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KEYWORD
| sign,tabl,changed
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 09 2007
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