login
A071947
Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.
3
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
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
KEYWORD
nonn,easy,tabl
AUTHOR
N. J. A. Sloane, Jun 15 2002
EXTENSIONS
Edited by Emeric Deutsch, Mar 04 2004
STATUS
approved