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A157905 Triangle read by rows, T(n,k) = A000055(n-k) * (A157904 * 0^(n-k)) 2
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 * A260931 A293819 A027113

Adjacent sequences:  A157902 A157903 A157904 * A157906 A157907 A157908

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Mar 08 2009

STATUS

approved

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Last modified May 9 05:06 EDT 2021. Contains 343688 sequences. (Running on oeis4.)