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

0,8

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.

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

Example

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

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 *)

Crossrefs

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

Cf. A027907.

Sequence in context: A265848 A139438 A135392 * A139343 A059247 A340940

Adjacent sequences: A071944 A071945 A071946 * A071948 A071949 A071950

Keyword

nonn,easy,tabl

Author

N. J. A. Sloane, Jun 15 2002

Extensions

Edited by Emeric Deutsch, Mar 04 2004

Status

approved