A157905 Triangle read by rows, T(n,k) = A000055(n-k) * (A157904 * 0^(n-k)).

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 2, 1, 2, 4, 8, 3, 2, 2, 4, 8, 17, 6, 3, 4, 4, 8, 17, 36, 11, 6, 6, 8, 8, 17, 36, 78, 23, 11, 12, 12, 16, 17, 36, 78, 170, 47, 23, 22, 24, 24, 34, 36, 78, 170, 375, 106, 47, 46, 44, 48, 51, 72, 78, 170, 375, 833

Offset

0,6

Comments

As a property of eigentriangles, sum of n-th row terms = rightmost term of next row.

Links

Table of n, a(n) for n=0..65.

Formula

Triangle read by rows, T(n,k) = A000055(n-k) * (A157904 * 0^(n-k)). A000055(n-k) = an infinite lower triangular matrix with A000055 in every column: (1, 1, 1, 1, 2, 3, 6, 11, 23, ...). (A157904 * 0^(n-k)) = a matrix with A157904 as the diagonal and the rest zeros.

Example

First few rows of the triangle =
    1;
    1,   1;
    1,   1,  2;
    1,   1,  2,  4;
    2,   1,  2,  4,  8;
    3,   2,  2,  4,  8,  17;
    6,   3,  4,  4,  8,  17,  36;
   11,   6,  6,  8,  8,  17,  36,  78;
   23,  11, 12, 12, 16,  17,  36,  78, 170;
   47,  23, 22, 24, 24,  34,  36,  78, 170, 375;
  106,  47, 46, 44, 48,  51,  72,  78, 170, 375, 833;
  235, 106, 94, 92, 88, 102, 108, 156, 170, 375, 833, 1870;
  ...
Row 5 = (3, 2, 2, 4, 8, 17) = termwise products of (3, 2, 1, 1, 1, 1) and (1, 1, 2, 4, 8, 17).

Mathematica

b[n_] := b[n] = If[n <= 1, n, Sum[Sum[d b[d], {d, Divisors[j]}] b[n - j], {j, 1, n - 1}]/(n - 1)];
t[n_] := t[n] = If[n == 0, 1, b[n] - (Sum[b[k] b[n - k], {k, 1, n - 1}] - If[OddQ[n], 0, b[n/2]])/2];
u[n_] := u[n] = If[n <= 0, 1, Sum[t[i] u[n - i - 1], {i, 0, n}]];
c[0] = 0; c[1] = 1; c[n_] := c[n] = Sum[d c[d] c[n - j], {j, 1, n - 1}, {d, Divisors[j]}]/(n - 1);
v[0] = 1; v[n_] := c[n] - (Sum[c[k] c[n - k], {k, 0, n}] - If[Mod[n, 2] == 0, c[n/2], 0])/2;
T[n_, k_] := v[n - k] u[k - 1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 21 2020, after Alois P. Heinz in A000055 and A157904 *)

Crossrefs

Cf. A000055 (first column), A157904 (row sums).

Sequence in context: A160266 A322134 A023504 * A356718 A260931 A293819

Adjacent sequences: A157902 A157903 A157904 * A157906 A157907 A157908

Keyword

nonn,tabl

Author

Gary W. Adamson, Mar 08 2009

Status

approved