A124800 Let M be a diagonal matrix with A007442 on the diagonal and P = Pascal's triangle as an infinite lower triangular matrix. Now read the triangle P*M by rows.

2, 2, 1, 2, 2, 1, 2, 3, 3, -1, 2, 4, 6, -4, 3, 2, 5, 10, -10, 15, -9, 2, 6, 15, -20, 45, -54, 23, 2, 7, 21, -35, 105, -189, 161, -53, 2, 8, 28, -56, 210, -504, 644, -424, 115, 2, 9, 36, -84, 378, -1134, 1932, -1908, 1035, -237, 2, 10, 45, -120, 630, -2268, 4830, -6360, 5175, -2370, 457

Offset: 1

Author

Gary W. Adamson, Nov 07 2006

Keywords

• tabl
• sign

Comments

Row sums = primes.

Right border = A007442, (2, 1, 1, -1, 3, -9...), = inverse binomial transform of the primes.

Examples

			Row 5: sum = 11 = p5 since (2 + 4 + 6 - 4 + 3) = 11.
Triangle begins:
  {2},
  {2, 1},
  {2, 2, 1},
  {2, 3, 3, -1},
  {2, 4, 6, -4, 3},
  {2, 5, 10, -10, 15, -9},
  {2, 6, 15, -20, 45, -54, 23},
  {2, 7, 21, -35, 105, -189, 161, -53},
  {2, 8, 28, -56, 210, -504, 644, -424, 115},
  {2, 9, 36, -84, 378, -1134, 1932, -1908, 1035, -237},
  {2, 10, 45, -120, 630, -2268, 4830, -6360, 5175, -2370, 457}
  ...
		

Crossrefs

Cf. A007442.

Programs

Mathematica

p[x_, n_] = Sum[Prime[k + 1]*Binomial[n, k]*x^k*(1 - x)^(n - k), {k, 0, n}];
Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}] // Flatten (* Roger L. Bagula and Gary W. Adamson, Oct 01 2008 *)

Formula

p(x,n) = Sum_{k=0..n} prime(k + 1) * binomial(n,k) * x^k * (1 - x)^(n - k); t(n,m) = coefficients(p(x,n)). - Roger L. Bagula and Gary W. Adamson, Oct 01 2008

Extensions

More terms from Roger L. Bagula and Gary W. Adamson, Oct 01 2008