cp's OEIS Frontend

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

%I A101491 #22 Jan 17 2025 16:24:55
%S A101491 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,
%T A101491 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,
%U A101491 210,120,120,45,45,10,10,1,1,286,462,330,330,165,165,55,55,11,11,1,1
%N A101491 Triangle T(n,k), read by rows: number of Knödel walks starting at 0, ending at k, with n steps.
%H A101491 Paolo Xausa, <a href="/A101491/b101491.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).
%H A101491 Helmut Prodinger, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s50proding.html">The Kernel Method: a collection of examples</a>, Séminaire Lotharingien de Combinatoire, B50f (2004), 19 pp.
%F A101491 G.f.: r(z)/(z*(1+z)*(1-r(z)))*(1+x*z*r(z))/(1-x*r(z)), with r(z) = (1-sqrt(1-4*z^2))/(2*z). Then the g.f. of the k-th column is r(z)^(k+1)/(z*(1-r(z))).
%F A101491 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)).
%e A101491 Triangle begins:
%e A101491   1,
%e A101491   0,1,
%e A101491   2,1,1,
%e A101491   1,3,1,1,
%e A101491   5,4,4,1,1,
%e A101491   5,10,5,5,1,1,
%e A101491   15,15,15,6,6,1,1,
%e A101491   20,35,21,21,7,7,1,1,
%e A101491   50,56,56,28,28,8,8,1,1,
%e A101491   76,126,84,84,36,36,9,9,1,1,
%e A101491   ...
%t A101491 A101491[n_, k_] := If[k == 0, Sum[(-1)^(n - i)*Binomial[i, BitShiftRight[i]], {i, 0, n}], Binomial[n, BitShiftRight[n - k]]];
%t A101491 Table[A101491[n, k], {n, 0, 15}, {k, 0, n}] (* _Paolo Xausa_, Jan 17 2025 *)
%o A101491 (PARI) T(n, k) = if (k==0, sum(i=0, n, (-1)^(n-i)*binomial(i, i\2)), binomial(n, (n-k)\2));
%o A101491 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print();); \\ _Michel Marcus_, Dec 04 2016
%Y A101491 Left-hand columns include A086905, A037952, A037955, A037951, A037956, A037953, A037957, A037954, A037958.
%K A101491 nonn,tabl
%O A101491 0,4
%A A101491 _Ralf Stephan_, Jan 21 2005