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A098473
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Triangle T(n,k) read by rows, T(n, k) = binomial(2*k, k)*binomial(n, k), 0<=k<=n.
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5
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1, 1, 2, 1, 4, 6, 1, 6, 18, 20, 1, 8, 36, 80, 70, 1, 10, 60, 200, 350, 252, 1, 12, 90, 400, 1050, 1512, 924, 1, 14, 126, 700, 2450, 5292, 6468, 3432, 1, 16, 168, 1120, 4900, 14112, 25872, 27456, 12870, 1, 18, 216, 1680, 8820, 31752, 77616, 123552, 115830
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OFFSET
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0,3
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COMMENTS
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This sequence gives the coefficients of the Jensen polynomials (increasing powers of x) of degree n and shift 0 for the central binomial sequence A000984. For a definition of Jensen polynomials see a comment in A094436. - Wolfdieter Lang, Jun 25 2019
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LINKS
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FORMULA
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T(n, k) = binomial(2*k, k)*binomial(n, k).
Sum_{k=0..n} T(n,k)*x^(n-k) = A126869(n), A002426(n), A000984(n), A026375(n), A081671(n), A098409(n), A098410(n) for x = -2, -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Sep 28 2007
O.g.f.: 1/sqrt(1 - t)*1/sqrt(1 - t*(1 + 4*x)) = 1 + (2*x + 1)*t + (1 + 4*x + 6*x^2)*t^2 + ....
Let R_n(x) denote the row generating polynomials of this triangle, which begin
R_1(x) = 1 + 2*x, R_2(x) = 1 + 4*x + 6*x^2, R_3(x) = 1 + 6*x + 18*x^2 + 20*x^3.
Dasbach gives the following slowly converging series for the logarithm function:
log(x) = Sum_{n >= 1} 1/n*R_n(-1/x), valid for x >= 4.
The polynomials (1 - x)^n*R_n(x/(1 - x)) appear to be the row polynomials of A135091 (see also A171128). (End)
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EXAMPLE
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Rows begin
1;
1, 2;
1, 4, 6;
1, 6, 18, 20;
1, 8, 36, 80, 70;
1, 10, 60, 200, 350, 252;
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MAPLE
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A098473 := proc(n, k) binomial(2*k, k)*binomial(n, k) ; end proc:
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MATHEMATICA
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Table[Binomial[2k, k]Binomial[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Aug 15 2020 *)
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PROG
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(PARI): T(n, k)=binomial(2*k, k)*binomial(n, k);
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print()); /* as triangle */
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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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