A101491 Triangle T(n,k), read by rows: number of Knödel walks starting at 0, ending at k, with n steps.

1, 0, 1, 2, 1, 1, 1, 3, 1, 1, 5, 4, 4, 1, 1, 5, 10, 5, 5, 1, 1, 15, 15, 15, 6, 6, 1, 1, 20, 35, 21, 21, 7, 7, 1, 1, 50, 56, 56, 28, 28, 8, 8, 1, 1, 76, 126, 84, 84, 36, 36, 9, 9, 1, 1, 176, 210, 210, 120, 120, 45, 45, 10, 10, 1, 1, 286, 462, 330, 330, 165, 165, 55, 55, 11, 11, 1, 1

Offset

0,4

Links

Table of n, a(n) for n=0..77.

H. Prodinger, The Kernel Method: a collection of examples, Séminaire Lotharingien de Combinatoire, B50f (2004), 19 pp.

Formula

G.f.: r(z)/(z(1+z)(1-r(z)))*(1+xzr(z))/(1-xr(z)), with r(z)=(1-sqrt(1-4z^2)/2z. Then the g.f. of the k-th column is r(z)^(k+1)/(z(1-r(z))).

T(n, k) = Sum_{i=0..n} (-1)^(n-i)*C(i, floor(i/2)) for k=0, otherwise T(n, k) = C(n, floor((n-k)/2)).

Example

1,
0,1,
2,1,1,
1,3,1,1,
5,4,4,1,1,
5,10,5,5,1,1,
15,15,15,6,6,1,1,
20,35,21,21,7,7,1,1,
50,56,56,28,28,8,8,1,1,
76,126,84,84,36,36,9,9,1,1,

Prog

(PARI) T(n, k) = if (k==0, sum(i=0, n, (-1)^(n-i)*binomial(i, i\2)), binomial(n, (n-k)\2));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print(); ); \\ Michel Marcus, Dec 04 2016

Crossrefs

Left-hand columns include A086905, A037952, A037955, A037951, A037956, A037953, A037957, A037954, A037958.

Sequence in context: A330794 A102054 A111604 * A276949 A205794 A241665

Adjacent sequences: A101488 A101489 A101490 * A101492 A101493 A101494

Keyword

nonn,tabl

Author

Ralf Stephan, Jan 21 2005

Status

approved