A143358 Triangle read by rows: T(n,k) = 2^k*binomial(n,k)*binomial(n-k, floor((n-k)/2)), 0 <= k <= n.

1, 1, 2, 2, 4, 4, 3, 12, 12, 8, 6, 24, 48, 32, 16, 10, 60, 120, 160, 80, 32, 20, 120, 360, 480, 480, 192, 64, 35, 280, 840, 1680, 1680, 1344, 448, 128, 70, 560, 2240, 4480, 6720, 5376, 3584, 1024, 256, 126, 1260, 5040, 13440, 20160, 24192, 16128, 9216, 2304, 512

Offset

0,3

Comments

Sum of terms in row n = binomial(2n+1,n) (A001700; see the Andreescu-Feng reference).

References

T. Andreescu and Z. Feng, 102 Combinatorial Problems (from the training of the USA IMO team), Birkhauser, Boston, 2003, Advanced problem # 15, pp. 11,61-63.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Formula

E.g.f.: exp(2*x*y)*(BesselI(0,2*x) + BesselI(1,2*x)). - Vladeta Jovovic, Dec 02 2008

Example

Triangle begins:
   1;
   1,  2;
   2,  4,   4;
   3, 12,  12,   8;
   6, 24,  48,  32, 16;
  10, 60, 120, 160, 80, 32;
  ...

Maple

T:=proc(n, k) options operator, arrow: 2^k*binomial(n, k)*binomial(n-k, floor((1/2)*n-(1/2)*k)) end proc: for n from 0 to 9 do seq(T(n, k), k=0..n) end do; # yields sequence in triangular form

Mathematica

Flatten[Table[2^k Binomial[n, k]Binomial[n-k, Floor[(n-k)/2]], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Jun 28 2021 *)

Crossrefs

Cf. A001700.

Sequence in context: A243238 A169629 A231731 * A006460 A064137 A329588

Adjacent sequences: A143355 A143356 A143357 * A143359 A143360 A143361

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 11 2008

Status

approved