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 A121576 Riordan array (2-2*x-sqrt(1-8*x+4*x^2), (1-2*x-sqrt(1-8*x+4*x^2))/2). 6
 1, 2, 1, 6, 5, 1, 24, 24, 8, 1, 114, 123, 51, 11, 1, 600, 672, 312, 87, 14, 1, 3372, 3858, 1914, 618, 132, 17, 1, 19824, 22992, 11904, 4218, 1068, 186, 20, 1, 120426, 140991, 75183, 28383, 8043, 1689, 249, 23, 1, 749976, 884112, 481704, 190347, 58398, 13929, 2508, 321, 26, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Inverse of Riordan array (1/(1+2*x), x*(1-x)/(1+2*x)). Row sums are A047891; first column is A054872. Signed version given by A121575. Triangle T(n,k), 0 <= k <= n, read by rows, given by [2, 1, 3, 1, 3, 1, 3, 1, 3, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 09 2006 LINKS G. C. Greubel, Rows n=0..100 of triangle, flattened FORMULA T(n,k) = [x^(n-k)](1-2*x-2*x^2)*(1+2*x)^n/(1-x)^(n+1) = (1/2)*Sum_{i=0..n-k} binomial(n,i) * binomial(2*n-k-i,n) * (4 - 9*i + 3*i^2 - 6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i. - Emanuele Munarini, May 18 2011 EXAMPLE Triangle begins 1; 2, 1; 6, 5, 1; 24, 24, 8, 1; 114, 123, 51, 11, 1; 600, 672, 312, 87, 14, 1; 3372, 3858, 1914, 618, 132, 17, 1; From Paul Barry, Apr 27 2009: (Start) Production matrix is 2, 1, 2, 3, 1, 2, 3, 3, 1, 2, 3, 3, 3, 1, 2, 3, 3, 3, 3, 1, 2, 3, 3, 3, 3, 3, 1, 2, 3, 3, 3, 3, 3, 3, 1 In general, the production matrix of the inverse of (1/(1-rx),x(1-x)/(1-rx)) is -r, 1, -r, 1 - r, 1, -r, 1 - r, 1 - r, 1, -r, 1 - r, 1 - r, 1 - r, 1, -r, 1 - r, 1 - r, 1 - r, 1 - r, 1, -r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1, -r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1 (End) MATHEMATICA Flatten[Table[Sum[Binomial[n, i]Binomial[2n-k-i, n](4-9i+3i^2-6(i-1)n+2n^2)/((n-i+2)(n-i+1))2^i, {i, 0, n-k}]/2, {n, 0, 8}, {k, 0, n}]] (* Emanuele Munarini, May 18 2011 *) PROG (Maxima) create_list(sum(binomial(n, i)*binomial(2*n-k-i, n)*(4-9*i+3*i^2-6*(i-1)*n+2*n^2)/((n-i+2)*(n-i+1))*2^i, i, 0, n-k)/2, n, 0, 8, k, 0, n); /* Emanuele Munarini, May 18 2011 */ (PARI) for(n=0, 10, for(k=0, n, print1(sum(j=0, n-k, 2^j*binomial(n, j) *binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1)))/2, ", "))) \\ G. C. Greubel, Nov 02 2018 (Magma) [[(&+[ 2^j*Binomial(n, j)*Binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1))/2: j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 02 2018 CROSSREFS Sequence in context: A179456 A214152 A121575 * A049444 A136124 A143491 Adjacent sequences: A121573 A121574 A121575 * A121577 A121578 A121579 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Aug 08 2006 STATUS approved

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Last modified August 8 06:45 EDT 2024. Contains 375020 sequences. (Running on oeis4.)