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.

1, 1, 2, 3, 4, 8, 5, 6, 8, 16, 35, 40, 48, 64, 128, 63, 70, 80, 96, 128, 256, 231, 252, 280, 320, 384, 512, 1024, 429, 462, 504, 560, 640, 768, 1024, 2048, 6435, 6864, 7392, 8064, 8960, 10240, 12288, 16384, 32768, 12155, 12870, 13728, 14784, 16128, 17920, 20480, 24576, 32768, 65536

Offset: 0

Author

Emeric Deutsch, Apr 12 2008

Keywords

• nonn
• tabl

Comments

The polynomials Q[n](x) arise in a contact problem in elasticity theory.

Row sums yield A001803.

T(n,0) = A001790(n).

T(n,n) = A046161(n).

Examples

			Triangle begins:
   1,
   1,  2,
   3,  4,  8,
   5,  6,  8, 16,
  35, 40, 48, 64, 128,
  63, 70, 80, 96, 128, 256,
  ...
		

References

• 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.

Crossrefs

Cf. A001803 (row sums), A001790 (1st column), A046161 (right diagonal).

Programs

Maple

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;

Mathematica

b[n_] := Numerator[Binomial[2n, n]/2^n];
Q[n_][x_] := HypergeometricPFQ[{-n, 1}, {1/2 - n}, x];
T[n_, k_] := Coefficient[b[n]*Q[n][x], x, k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 23 2024 *)

Formula

Q[n](x) = (2n+1)*(Integral_{t=0..sqrt(1-x)} (x+t^2)^n dt)/sqrt(1-x).

Q[n](x) = 1 + 2*n*x*Q[n-1](x)/(2n-1).