A062135 Odd-numbered columns of Losanitsch triangle A034851 formatted as triangle with an additional first column.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 6, 3, 1, 0, 3, 10, 12, 4, 1, 0, 3, 19, 28, 20, 5, 1, 0, 4, 28, 66, 60, 30, 6, 1, 0, 4, 44, 126, 170, 110, 42, 7, 1, 0, 5, 60, 236, 396, 365, 182, 56, 8, 1, 0, 5, 85, 396, 868, 1001, 693, 280, 72, 9, 1

Offset

0,8

Comments

Because the sequence of column m=2*k, k >= 1, of A034851 is the partial sum sequence of the one of column m=2*k-1 the present triangle is essentially Losanitsch's triangle A034851.

Row sums give A051450 with A051450(0) := 1. Column sequences (without leading zeros) are for m=0..6: A000007, A008619, A005993, A005995, A018211, A018213, A062136.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened)

Formula

T(n, m) = A034851(n-1+m, n-m), n >= m >= 0; A034851(n-1, n) := 0, n >= 1, A034851(-1, 0) := 1.

T(n, m) = 0 if n<m; T(0, 0)=1, T(n, 0)=0 if n >= 1; T(n, m) = T(n-1, m)+sum(T(k, m-1), k=m-1..n-1) if n+m even and T(n, m) = T(n-1, m)+sum(T(k, m-1), k=m-1..n-1)-binomial((n+m-3)/2, m-1) if n+m odd, n >= m >= 1.

G.f. for column m: x^m*Pe(m, x^2)/(((1-x)^(2*m))*(1+x)^m), m >= 0, with Pe(m, x^2)= sum(A034839(m, k)*x^(2*k), k=0..floor(n/2)), the row polynomial of array A034839 (even-indexed entries of the rows of Pascal's triangle).

Example

Triangle begins:
  {1};
  {0,1};
  {0,1,1};
  {0,2,2,1};
  ...
Pe(4,x^2)=1+6*x^2+x^4.

Mathematica

t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2; t[n_, k_] := (Binomial[n, k] + Binomial[Quotient[n, 2], Quotient[k, 2]])/2; Flatten[Table[t[n - 1 + m, n - m], {n, 0, 12}, {m, 0, n}]] (* Michael De Vlieger, Sep 28 2024, after Jean-François Alcover at A034851  *)

Crossrefs

Cf. A034851, A034839.

Sequence in context: A177975 A340995 A363733 * A190182 A371788 A377730

Adjacent sequences: A062132 A062133 A062134 * A062136 A062137 A062138

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Jun 19 2001

Extensions

More terms from Michael De Vlieger, Sep 28 2024

Status

approved