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A138773 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial P[n](x) = b(n)Q[n](x), where b(n) = numerator of binomial(2n,n)/2^n = A001790(n) and Q[n](x) = F(-n,1; 1/2-n; x) (hypergeometric function); 0 <= k <= n.

%I A138773 #12 Jul 23 2024 08:59:22
%S A138773 1,1,2,3,4,8,5,6,8,16,35,40,48,64,128,63,70,80,96,128,256,231,252,280,
%T A138773 320,384,512,1024,429,462,504,560,640,768,1024,2048,6435,6864,7392,
%U A138773 8064,8960,10240,12288,16384,32768,12155,12870,13728,14784,16128,17920,20480,24576,32768,65536
%N A138773 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial P[n](x) = b(n)Q[n](x), where b(n) = numerator of binomial(2n,n)/2^n = A001790(n) and Q[n](x) = F(-n,1; 1/2-n; x) (hypergeometric function); 0 <= k <= n.
%C A138773 The polynomials Q[n](x) arise in a contact problem in elasticity theory.
%C A138773 Row sums yield A001803.
%C A138773 T(n,0) = A001790(n).
%C A138773 T(n,n) = A046161(n).
%D A138773 E. G. Deich (E. Deutsch), On an axially symmetric contact problem for a non-plane stamp with a circular cross-section (in Russian), Prikl. Mat. Mekh., 26, No. 5, 1962, 931-934.
%F A138773 Q[n](x) = (2n+1)*(Integral_{t=0..sqrt(1-x)} (x+t^2)^n dt)/sqrt(1-x).
%F A138773 Q[n](x) = 1 + 2*n*x*Q[n-1](x)/(2n-1).
%e A138773 Triangle begins:
%e A138773    1,
%e A138773    1,  2,
%e A138773    3,  4,  8,
%e A138773    5,  6,  8, 16,
%e A138773   35, 40, 48, 64, 128,
%e A138773   63, 70, 80, 96, 128, 256,
%e A138773   ...
%p A138773 p:=proc(n) options operator, arrow: numer(simplify(hypergeom([ -n, 1], [1/2-n], x))) end proc: for n from 0 to 9 do P[n]:=p(n) end do: for n from 0 to 9 do seq(coeff(P[n],x,k),k=0..n) end do;
%t A138773 b[n_] := Numerator[Binomial[2n, n]/2^n];
%t A138773 Q[n_][x_] := HypergeometricPFQ[{-n, 1}, {1/2 - n}, x];
%t A138773 T[n_, k_] := Coefficient[b[n]*Q[n][x], x, k];
%t A138773 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 23 2024 *)
%Y A138773 Cf. A001803 (row sums), A001790 (1st column), A046161 (right diagonal).
%K A138773 nonn,tabl
%O A138773 0,3
%A A138773 _Emeric Deutsch_, Apr 12 2008