A037954 -id:A037954 - 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.

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

Ralf Stephan, Jan 21 2005

			Triangle begins:
  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,
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Helmut Prodinger, The Kernel Method: a collection of examples, Séminaire Lotharingien de Combinatoire, B50f (2004), 19 pp.

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

A101491[n_, k_] := If[k == 0, Sum[(-1)^(n - i)*Binomial[i, BitShiftRight[i]], {i, 0, n}], Binomial[n, BitShiftRight[n - k]]];
Table[A101491[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Jan 17 2025 *)

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

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))).

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)).

A335322 Triangle read by rows: T(n, k) = binomial(n, floor((n+k+1)/2)) with k <= n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 10, 5, 5, 1, 1, 15, 15, 6, 6, 1, 1, 35, 21, 21, 7, 7, 1, 1, 56, 56, 28, 28, 8, 8, 1, 1, 126, 84, 84, 36, 36, 9, 9, 1, 1, 210, 210, 120, 120, 45, 45, 10, 10, 1, 1, 462, 330, 330, 165, 165, 55, 55, 11, 11, 1, 1, 792, 792, 495, 495, 220, 220, 66, 66, 12, 12, 1, 1

Offset: 1

Stefano Spezia, May 31 2020

T(n, k) is a tight upper bound of the cardinality of an intersecting Sperner family or antichain of the set {1, 2,..., n}, where every collection of pairwise independent subsets is characterized by an intersection of cardinality at least k (see Theorem 1.3 in Wong and Tay).

Equals A061554 with the first row of the array (resp. the first column of the triangle) removed. - Georg Fischer, Jul 26 2023

			The triangle T(n, k) begins
n\k|  1   2   3   4   5   6   7   8
---+-------------------------------
1  |  1
2  |  1   1
3  |  3   1   1
4  |  4   4   1   1
5  | 10   5   5   1   1
6  | 15  15   6   6   1   1
7  | 35  21  21   7   7   1   1
8  | 56  56  28  28   8   8   1   1
...

Eric Charles Milner, A Combinatorial Theorem On Systems of Sets, Journal of the London Mathematical Society, 43, (1968), 204-206.
W. H. W. Wong and E. G. Tay, On Cross-intersecting Sperner Families, arXiv:2001.01910 [math.CO], 2020.
Index entries for sequences related to antichains.

Cf. A000372, A001405, A004526, A007318, A007695, A061554, A266696, A325982, A325983.

Cf. A037951 (k=3), A037952 (k=1), A037953 (k=5), A037954 (k=7), A037955 (k=2), A037956 (k=4), A037957 (k=6), A037958 (k=8), A045621 (row sums).

T[n_,k_]:=Binomial[n,Floor[(n+k+1)/2]]; Table[T[n,k],{n,12},{k,n}]//Flatten

T(n, k) = binomial(n, (n+k+1)\2);
vector(10, n, vector(n, k, T(n, k))) \\ Michel Marcus, Jun 01 2020

T(n, k) = A007318(n, A004526(n+k+1)) with k <= n.

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.

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