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

0,3

Comments

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

Links

Table of n, a(n) for n=0..54.

E. G. Deich (Э. Г. Дейч; Emeric 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.

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).

Row sums yield A001803.

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

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

From Thomas Scheuerle, Oct 13 2025: (Start)

T(n, k) = binomial(2*n-2*m, n-m)*n!/(2^(n-2*m)*denominator(2^n/n!)).

This triangle can be used to construct the series expansion of the elliptic integral of the third kind: Integral_{0..Pi/2} 1/(sqrt(1-m*sin(t)^2)*(1-k*sin(t)^2)) dt = Pi*(1/(2*sqrt(1-k))) - Sum_{j>=1} m^j*A001790(j)*( 1/(2*A161737(j+1)) ) *( (Sum_{r=1..j-1} T(j-1, r-1)*k^(-r)) + T(j-1, j-1)*(1-1/sqrt(1-k))*k^(-j) ). (End)

Example

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

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 *)

Prog

(PARI) T(n, k) = binomial(2*n-2*m, n-m)*n!/(2^(n-2*m)*denominator(2^n/n!)) \\Thomas Scheuerle, Oct 13 2025

Crossrefs

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

Cf. A161737.

Sequence in context: A229597 A366656 A175060 * A132989 A283814 A360597

Adjacent sequences: A138770 A138771 A138772 * A138774 A138775 A138776

Keyword

nonn,tabl

Author

Emeric Deutsch, Apr 12 2008

Status

approved