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

0,5

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

Links

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A195284 A351263 A076341 * A337506 A327363 A051122

Adjacent sequences: A110507 A110508 A110509 * A110511 A110512 A110513

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 24 2005

Status

approved