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A098473 Triangle T(n,k) read by rows, T(n, k) = binomial(2*k, k)*binomial(n, k), 0<=k<=n.

%I A098473 #40 Mar 18 2025 16:20:44
%S A098473 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,
%T A098473 1050,1512,924,1,14,126,700,2450,5292,6468,3432,1,16,168,1120,4900,
%U A098473 14112,25872,27456,12870,1,18,216,1680,8820,31752,77616,123552,115830
%N A098473 Triangle T(n,k) read by rows, T(n, k) = binomial(2*k, k)*binomial(n, k), 0<=k<=n.
%C A098473 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
%H A098473 Indranil Ghosh, <a href="/A098473/b098473.txt">Rows 0..125, flattened</a>
%H A098473 O. T. Dasbach, <a href="https://doi.org/10.37236/880">A natural series for the natural logarithm</a>, Electronic Journal of Combinatorics, (15) 2008 #N5.
%F A098473 T(n, k) = binomial(2*k, k)*binomial(n, k).
%F A098473 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
%F A098473 From _Peter Bala_, Jun 06 2011: (Start)
%F A098473 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 + ....
%F A098473 Let R_n(x) denote the row generating polynomials of this triangle, which begin
%F A098473 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.
%F A098473 Dasbach gives the following slowly converging series for the logarithm function:
%F A098473 log(x)  = Sum_{n >= 1} 1/n*R_n(-1/x), valid for x >= 4.
%F A098473 The polynomials (1 - x)^n*R_n(x/(1 - x)) appear to be the row polynomials of A135091 (see also A171128). (End)
%e A098473 Rows begin
%e A098473   1;
%e A098473   1,  2;
%e A098473   1,  4,  6;
%e A098473   1,  6, 18,  20;
%e A098473   1,  8, 36,  80,  70;
%e A098473   1, 10, 60, 200, 350, 252;
%p A098473 A098473 := proc(n,k) binomial(2*k,k)*binomial(n,k) ; end proc:
%t A098473 Table[Binomial[2k,k]Binomial[n,k],{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Aug 15 2020 *)
%o A098473 (PARI) T(n,k)=binomial(2*k, k)*binomial(n, k);
%o A098473 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()); /* as triangle */
%Y A098473 Row sums are A026375.
%Y A098473 Antidiagonal sums are A026569.
%Y A098473 Principal diagonal is A000984.
%Y A098473 Cf. A002426, A081671, A098409, A098410, A094436, A126869.
%K A098473 easy,nonn,tabl
%O A098473 0,3
%A A098473 _Paul Barry_, Sep 09 2004