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Search: id:A104505
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| A104505 |
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Triangle, read by rows, equal to the right-hand side of the triangle A084610, with row n listing the coefficients of (1+x-x^2)^n: T(n,k) = [x^(n+k)] (1+x-x^2)^n, for n>=k>=0. |
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+0
6
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| 1, 1, -1, -1, -2, 1, -5, 0, 3, -1, -5, 8, 2, -4, 1, 11, 15, -10, -5, 5, -1, 41, -6, -30, 10, 9, -6, 1, 29, -77, -14, 49, -7, -14, 7, -1, -125, -120, 112, 56, -70, 0, 20, -8, 1, -365, 117, 288, -126, -126, 90, 12, -27, 9, -1, -131, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1, 1409, 946, -1265, -495, 858, 33, -363, 110, 55, -44
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Matrix inverse is triangle A104509 and is related to Fibonacci numbers. Column 0 equals A098331, with g.f.: 1/sqrt(1-2*x+5*x^2). Column 1 equals A104506, with g.f.: ((1-x)/sqrt(1-2*x+5*x^2)-1)/(2*x). Row sums equal A104507. Absolute row sums equal A104508.
Array (1/sqrt(1-2x+5x^2), (1-x-sqrt(1-2x+5x^2))/(2x)), in Riordan array notation. Product of A120616 by A007318. Column k has e.g.f. exp(x)Bessel_I(k,2*sqrt(-1)x)*(sqrt(-1))^k. - Paul Barry (pbarry(AT)wit.ie), Jun 17 2006
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FORMULA
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T(n, 0) = A098331(n). T(n, 1) = n*A007440(n) (n>0).
Column k has e.g.f. exp(x)Bessel_I(k,2*sqrt(-1)x)*(sqrt(-1))^k - Paul Barry (pbarry(AT)wit.ie), Jun 17 2006
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EXAMPLE
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Rows begin:
1;
1,-1;
-1,-2,1;
-5,0,3,-1;
-5,8,2,-4,1;
11,15,-10,-5,5,-1;
41,-6,-30,10,9,-6,1;
29,-77,-14,49,-7,-14,7,-1;
-125,-120,112,56,-70,0,20,-8,1;
-365,117,288,-126,-126,90,12,-27,9,-1;
-131,770,45,-540,90,228,-105,-30,35,-10,1; ...
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PROGRAM
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(PARI) T(n, k)=if(n<k|k<0, 0, polcoeff((1+x-x^2)^n, n+k, x))
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CROSSREFS
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Cf. A104509, A098331, A007440, A104506, A104507, A104508.
Adjacent sequences: A104502 A104503 A104504 this_sequence A104506 A104507 A104508
Sequence in context: A053374 A093876 A127477 this_sequence A021469 A090985 A011131
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KEYWORD
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sign,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Mar 11 2005
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