A068914 -id:A068914 - cp's OEIS frontend

A334683 Antidiagonal sums of A068914.

1, 1, 2, 3, 5, 7, 12, 17, 30, 44, 79, 120, 218, 341, 625, 1001, 1848, 3016, 5603, 9283, 17342, 29085, 54606, 92513, 174456, 298111, 564333, 971563, 1845364, 3198130, 6092038, 10621153, 20282471, 35554559, 68042537, 119874494, 229840130, 406794862, 781246800, 1388643835

Offset: 0

Stefano Spezia, May 08 2020

nonn

Cf. A068914.

T[n_,k_]:=Sum[(-1)^j*Binomial[n,Floor[(n+(k+2)j)/2]],{j,Floor[-n/(k+2)],Ceiling[n/(k+2)]}]; Table[Sum[T[n-k,k],{k,0,n}],{n,0,39}]

T(n,k) = sum(j=floor(-n/(k+2)), ceil(n/(k+2)), (-1)^j*binomial(n,floor((n+(k+2)*j)/2))); \\ A068914
a(n) = sum(k=0, n, T(n-k, k)); \\ Michel Marcus, May 09 2020

A216226 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=4, T(0,0) = T(0,1) = T(0,2) = T(0,3) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 0, 3, 2, 0, 0, 0, 3, 5, 0, 0, 0, 0, 0, 8, 5, 0, 0, 0, 0, 0, 8, 13, 0, 0, 0, 0, 0, 0, 0, 21, 13, 0, 0, 0, 0, 0, 0, 0, 21, 34, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 34, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 13 2013

			Square array begins:
1, 1, 1, 1,  0,  0,  0,   0,   0, 0, ... row n=0
0, 1, 2, 3,  3,  0,  0,   0,   0, 0, ... row n=1
0, 0, 2, 5,  8,  8,  0,   0,   0, 0, ... row n=2
0, 0, 0, 5, 13, 21, 21,   0,   0, 0, ... row n=3
0, 0, 0, 0, 13, 34, 55,  55,   0, 0, ... row n=4
0, 0, 0, 0,  0, 34, 89, 144, 144, 0, ... row n=5
...

Cf. A000045 (Fibonacci numbers), A000285, A001519, A001906, A068914

Cf. Similar sequences: A216201, A216210, A216216, A216218, A216219, A216220, A216228, A216229, A216230

T(n,n) = A000045(2*n-1) = A001519(n).
T(n,n+1) = A000045(2*n+1) = A001519(n+1).
T(n,n+2) = T(n,n+3) = A000045(2*n+2) = A001906(n+1).
Sum_{k, 0<=k<=n} T(n-k,k) = A000045(n+1).
Sum_{k, k>=0} T(n,k) = A000285(2*n+1).
Sum_{n, n>=0} T(n,k) = A000285(2*k-2), k>=2.

A216228 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=3, T(0,0) = T(0,1) = T(0,2) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 32

Offset: 0

Philippe Deléham, Mar 13 2013

An arithmetic hexagon of E. Lucas.

			Square array begins:
1, 1, 1, 0, 0,  0,  0,  0, ... row n=0
0, 1, 2, 2, 0,  0,  0,  0, ... row n=1
0, 0, 2, 4, 4,  0,  0,  0, ... row n=2
0, 0, 0, 4, 8,  8,  0,  0, ... row n=3
0, 0, 0, 0, 8, 16, 16,  0, ... row n=4
0, 0, 0, 0, 0, 16, 32, 32, ... row n=5
...

E. Lucas, Théorie des nombres, Albert Blanchard, Paris 1958, Tome 1, p.89

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, 1991, p.89

Cf. A000079, A011782, A016116, A068914, A084215.

T(n,n) = A011782(n).
T(n,n+1) = T(n,n+2) = 2^n = A000079(n).
Sum_{k, 0<=k<=n} T(n-k,k) = A016116(n).
Sum_{n, n>=0} T(n,k) = A084215(k).
Sum_{k, k>=0} T(n,k) = A084215(n+1), n>=1.

A216230 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=2, T(0,0) = T(0,1) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 14 2013

			Square array begins:
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, ...
...

Cf. A000012, A068914

Cf. Similar sequences: A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229

T(n,n) = T(n,n+1) = 1.

Sum_{k, 0<=k<=n} T(n-k, k) = 1.

A132890 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n that have height k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 1, 7, 5, 5, 1, 1, 1, 7, 13, 6, 6, 1, 1, 1, 15, 18, 20, 7, 7, 1, 1, 1, 15, 39, 26, 27, 8, 8, 1, 1, 1, 31, 57, 73, 35, 35, 9, 9, 1, 1, 1, 31, 112, 99, 109, 44, 44, 10, 10, 1, 1, 1, 63, 169, 253, 152, 154, 54, 54, 11, 11, 1, 1

Offset: 1

Emeric Deutsch, Sep 08 2007

Sum of terms in row n = binomial(n, floor(n/2)) = A001405(n).
T(n,2) = A052551(n-2) (n >= 2).
T(n,3) = A005672(n) = Fibonacci(n+1) - 2^floor(n/2).
Sum_{k=1..n} k*T(n,k) = A132891(n).

			T(5,3)=4 because we have UDUUU, UUDUU, UUUDD and UUUDU, where U=(1,1) and D=(1,-1).
Triangle starts:
  1;
  1, 1;
  1, 1, 1;
  1, 3, 1, 1;
  1, 3, 4, 1; 1;
  1, 7, 5, 5, 1, 1;

Alois P. Heinz, Rows n = 1..141, flattened
Steven R. Finch, How far might we walk at random?, arXiv:1802.04615 [math.HO], 2018.
R. Kemp, On the average depth of a prefix of the Dycklanguage D_1, Discrete Math., 36, 1981, 155-170.
Toufik Mansour, Gokhan Yilidirim, Longest increasing subsequences in involutions avoiding patterns of length three, Turkish Journal of Mathematics (2019), Section 2.2

Cf. A001405, A052551, A005672, A132891, A068914.

v := ((1-sqrt(1-4*z^2))*1/2)/z: g := proc (k) options operator, arrow: v^k*(1+v)*(1+v^2)/((1+v^(k+1))*(1+v^(k+2))) end proc: T := proc (n, k) options operator, arrow: coeff(series(g(k), z = 0, 50), z, n) end proc: for n from 0 to 12 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, k) option remember; `if`(x=0, z^k, `if`(y>0,
      b(x-2, y-1, k), 0)+ b(x-2, y+1, max(y+1, k)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n))(b(2*n, 0$2)):
seq(T(n), n=1..16);  # Alois P. Heinz, Sep 05 2017

b[x_, y_, k_] := b[x, y, k] = If[x == 0, z^k, If[y > 0, b[x - 2, y - 1, k], 0] + b[x - 2, y + 1, Max[y + 1, k]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 1, n}]][b[2n, 0, 0]];
Table[T[n], {n, 1, 16}] // Flatten (* Jean-François Alcover, Apr 01 2018, after Alois P. Heinz *)

The g.f. of column k is g(k, z) = v^k*(1+v)*(1+v^2)*/((1+v^(k+1))*(1+v^(k+2))), where v = (1-sqrt(1-4*z^2))/(2*z). (Obtained as the difference G(k,z)-G(k-1,z), where G(k,z) is given in the R. Kemp reference (p. 159).)

Keyword tabl added by Michel Marcus, Apr 09 2013

A069713 As a square array T(n,k) by antidiagonals, number of ways of partitioning k into up to n parts each no more than 5, or into up to 5 parts each no more than n; as a triangle t(n,k), number of ways of partitioning n into exactly k parts each no more than 6 (i.e., of arranging k indistinguishable standard dice to produce a total of n).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 0, 3, 4, 3, 2, 1, 1, 0, 0, 3, 5, 5, 3, 2, 1, 1, 0, 0, 2, 6, 6, 5, 3, 2, 1, 1, 0, 0, 2, 6, 8, 7, 5, 3, 2, 1, 1, 0, 0, 1, 6, 9, 9, 7, 5, 3, 2, 1, 1, 0, 0, 1, 6, 11, 11, 10, 7, 5, 3, 2, 1, 1, 0, 0, 0, 5, 11, 14, 12, 10, 7, 5

Offset: 0

Henry Bottomley, Apr 01 2002

			As square array, rows start: 1,0,0,0,0,0,...; 1,1,1,1,1,1,...; 1,1,2,2,3,3,...; 1,1,2,3,4,5,...; 1,1,2,3,5,6,...; 1,1,2,3,5,7,...; etc. As triangle, rows start: 1; 0,1; 0,1,1; 0,1,1,1; 0,1,2,1,1; 0,1,2,2,1,1; 0,1,3,3,2,1,1; etc. T(3,7)=6 since 7 can be written as 5+2, 5+1+1, 4+3, 4+2+1, 3+3+1, 3+2+2; or alternatively as 2+2+1+1+1, 3+1+1+1, 2+2+2+1, 3+2+1+1, 3+2+2, 3+3+1. t(10,3)=6 since 10 can be written as 6+3+1, 6+2+2, 5+4+1, 5+3+2, 4+4+2, 4+3+3.

Cf. A061676 for a similar triangle, though with distinguishable dice (and a different offset). Antidiagonal sums of T(n, k), i.e., row sums (over k) of t(n, k), are A001402. First 22 terms are same as A068914 (see formula).

If k<6 T(n,k) = A068914(n,k). T(n,k) = T(n,5n-k); t(n,k) = t(7n-k,k). T(floor(5n/2),n) = t(n,floor(7n/2)) = A001975(n).

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A334683 Antidiagonal sums of A068914.

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A216226 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=4, T(0,0) = T(0,1) = T(0,2) = T(0,3) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A216228 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=3, T(0,0) = T(0,1) = T(0,2) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A216230 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=2, T(0,0) = T(0,1) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A132890 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n that have height k (1 <= k <= n).

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