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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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5
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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
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OFFSET
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0,5
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COMMENTS
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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. - Paul Barry, Jun 17 2006
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LINKS
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FORMULA
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Column k has e.g.f. exp(x)*Bessel_I(k,2*sqrt(-1)x)*(sqrt(-1))^k. - Paul Barry, Jun 17 2006
Matrix factorization in the Riordan group: ( 1/(1 - x), x/(1 - x) ) * ( 1/sqrt(1 + 4*x^2), (1 - sqrt(1 + 4*x^2))/(2*x) ) = A007318 * signed version of A108044.
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = (1 - x - sqrt(1 - 2*x + 5*x^2))/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = x^2 + x - 1. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)
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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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MATHEMATICA
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T[n_, k_] := Coefficient[(1 + x - x^2)^n, x, n + k];
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PROG
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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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KEYWORD
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AUTHOR
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STATUS
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approved
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