A110510 Riordan array (1, x*c(2x)), c(x) the g.f. of A000108.

1, 0, 1, 0, 2, 1, 0, 8, 4, 1, 0, 40, 20, 6, 1, 0, 224, 112, 36, 8, 1, 0, 1344, 672, 224, 56, 10, 1, 0, 8448, 4224, 1440, 384, 80, 12, 1, 0, 54912, 27456, 9504, 2640, 600, 108, 14, 1, 0, 366080, 183040, 64064, 18304, 4400, 880, 140, 16, 1, 0, 2489344, 1244672, 439296

Offset: 0

Author

Paul Barry, Jul 24 2005

Keywords

• easy
• nonn
• tabl

Comments

Row sums are C(2;n), A064062. Inverse is A110509. Diagonal sums are A108308. [Corrected by Philippe Deléham, Nov 09 2007]

Triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 2, 2, 2, 2, 2, 2, 2, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 23 2014

Examples

			Rows begin
  1;
  0,   1;
  0,   2,   1;
  0,   8,   4,   1;
  0,  40,  20,   6,   1;
  0, 224, 112,  36,   8,   1;
  ...
Production matrix begins:
  0,  1;
  0,  2,  1;
  0,  4,  2,  1;
  0,  8,  4,  2,  1;
  0, 16,  8,  4,  2,  1;
  0, 32, 16,  8,  4,  2,  1;
  0, 64, 32, 16,  8,  4,  2,  1;
  ... - _Philippe Deléham_, Sep 23 2014
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Programs

Mathematica

T[n_, k_] := (k/n)*Binomial[2*n - k - 1, n - k]*2^(n - k); Join[{1}, Table[T[n, k], {n, 1, 10}, {k, 0, n}]] // Flatten (* G. C. Greubel, Aug 29 2017 *)

PARI

concat([1], for(n=1,25, for(k=0,n, print1((k/n)*binomial(2*n-k-1, n-k)*2^(n-k), ", ")))) \\ G. C. Greubel, Aug 29 2017

Formula

Number triangle: T(0,k) = 0^k, T(n,k) = (k/n)*C(2n-k-1, n-k)*2^(n-k), n, k > 0.

T(n,k) = A106566(n,k)*2^(n-k). - Philippe Deléham, Nov 08 2007

T(n,k) = 2*T(n,k+1) + T(n-1,k-1) with T(n,n) = 1 and T(n,0) = 0 for n >= 1. - Peter Bala, Feb 02 2020