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 A113139 Number triangle, equal to half of Delannoy square array A008288. 8
 1, 3, 1, 13, 5, 1, 63, 25, 7, 1, 321, 129, 41, 9, 1, 1683, 681, 231, 61, 11, 1, 8989, 3653, 1289, 377, 85, 13, 1, 48639, 19825, 7183, 2241, 575, 113, 15, 1, 265729, 108545, 40081, 13073, 3649, 833, 145, 17, 1, 1462563, 598417, 224143, 75517, 22363, 5641 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Row sums are A047781(n+1). Diagonal sums are A113140. Inverse is A113141. LINKS Peter Bala, Notes on generalized Riordan arrays Peter Bala, A 4-parameter family of embedded Riordan arrays FORMULA T(n, k) = Sum_{j=0..n} C(n-k, j)*C(n+j, k+j). T(n, k) = Sum_{j=0..n} C(n, j)*C(n-k, j-k)*2^(n-j). From Peter Bala, Dec 09 2015: (Start) T(n,k) = A008288(n - k, n). O.g.f.: 2/( sqrt(x^2 - 6*x + 1)*(t*sqrt(x^2 - 6*x + 1) + t*x - t + 2) ) = 1 + (3 + t)*x + (13 + 5*t + t^2)*x^2 + .... Riordan array (f(x), x*g(x)), where f(x) = 1/sqrt(1 - 6*x + x^2) is the o.g.f. for the central Delannoy numbers, A001850, and g(x) = 1/x* revert( x*(1 - x)/(1 + x) ) = 1 + 2*x + 6*x^2 + 22*x^3 + 90*x^4 + 394*x^5 + ... is the o.g.f. for the large Schroder numbers, A006318. Read as a square array, this is the generalized Riordan array (f(x), g(x)) in the sense of the Bala link, which factorizes as (1 + x*g'(x)/g(x), x*g(x)) * (1/(1 - x), (1 + x)/(1 - x)) = A110171 * A008288. See the example below. (End) T(n,k) = (-1)^(n-k)*hypergeom([n+1, -n+k], [1], 2). - Peter Luschny, Mar 02 2017 From Peter Bala, Feb 16 2020: (Start) T(n,k) = P(n-k, k, 0, 3), where P(n, alpha, beta, x) is the n-th Jacobi polynomial with parameters alpha and beta. T(n,k) = binomial(n,k) * hypergeom( [n + 1, k - n], [k + 1], -1 ). The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 + x)^n/(1 - x)^(n+1) about 0. For example, for n = 4, (1 + x)^4/(1 - x)^5 = 1 + 9*x + 41*x^2 + 129*x^3 + 321*x^4 + O(x^5). Cf. A110171. (End) EXAMPLE Triangle begins      1;      3,    1;     13,    5,    1;     63,   25,    7,   1;    321,  129,   41,   9,  1;   1683,  681,  231,  61, 11,  1;   8989, 3653, 1289, 377, 85, 13, 1;   ... A113139 as a square array = A110171 * A008288:   / 1   1   1   1 ... \   / 1         \ / 1 1  1  1 ...\   | 3   5   7   9 ... |   | 2  1       || 1 3  5  7 ...|   |13  25  41  61 ... | = | 8  4 1     || 1 5 13 25 ...|   |63 129 231 377 ... |   |38 18 6 1   || 1 7 25 63 .. |   |...                |   |...         || 1...         | - Peter Bala, Dec 09 2015 MAPLE T := (n, k) -> (-1)^(n-k)*hypergeom([n+1, -n+k], [1], 2): seq(seq(simplify(T(n, k)), k=0..n), n=0..8); # Peter Luschny, Mar 02 2017 MATHEMATICA Table[Sum[Binomial[n - k, j] Binomial[n + j, k + j], {j, 0, n}], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 09 2015 *) CROSSREFS A001850 (column 0), A002002 (column 1), A026002 (column 2), A190666 (column 3), A047781 (row sums), A113140 (diagonal sums), A113141 (matrix inverse). Cf. A006318, A008288, A110171. Sequence in context: A277197 A297898 A322384 * A266577 A143411 A096773 Adjacent sequences:  A113136 A113137 A113138 * A113140 A113141 A113142 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Oct 15 2005 STATUS approved

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Last modified June 3 05:29 EDT 2020. Contains 334798 sequences. (Running on oeis4.)