A071947 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.

1, 1, 0, 1, 1, 1, 1, 2, 3, 1, 1, 3, 6, 6, 3, 1, 4, 10, 15, 15, 6, 1, 5, 15, 29, 40, 36, 15, 1, 6, 21, 49, 84, 105, 91, 36, 1, 7, 28, 76, 154, 238, 280, 232, 91, 1, 8, 36, 111, 258, 468, 672, 750, 603, 232, 1, 9, 45, 155, 405, 837, 1398, 1890, 2025, 1585, 603, 1, 10, 55, 209, 605

Offset: 0

Author

N. J. A. Sloane, Jun 15 2002

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins
  1;
  1,  0;
  1,  1,  1;
  1,  2,  3,  1;
  1,  3,  6,  6,  3;
  1,  4, 10, 15, 15,  6;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

Crossrefs

Row sums give A002426 (central trinomial coefficients). Reversal of A089942.

Cf. A027907.

Programs

Maple

A071947_row := proc(n) local G, k; G := expand((1+x+x^2)^n):
seq(coeff(G,x,k) - coeff(G,x,k-1), k=0..n) end:
seq(print(A071947_row(n)), n=0..11); # Peter Luschny, Oct 01 2014

Mathematica

A027907[n_, k_] := Sum[Binomial[n, j]*Binomial[j, k - j], {j, 0, n}]; A005043[n_] := Sum[(-1)^k*Binomial[n, k]*Binomial[k, Floor[k/2]], {k, 0, n}]; T[n_, k_] := A027907[n, k] - A027907[n, k - 1]; T[n_, n_] := A005043[n]; Table[T[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Mar 02 2017 *)

Formula

G.f.: t*(1+t*z-q)/[(1+t*z)*(2*t^2*z +t*z - 1 + q)], where q = sqrt(1 -2*t*z -3*t^2*z^2).

Sum_{k, 0<=k<=n} T(n,k)*2^(n-k) = A112657(n). - Philippe Deléham, Apr 01 2007

T(n,k) = A027907(n,k) - A027907(n,k-1). T(n,n) = A005043(n). # Peter Luschny, Oct 01 2014

Extensions

Edited by Emeric Deutsch, Mar 04 2004