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Triangle read by rows, T(n,k) = A000055(n-k) * (A157904 * 0^(n-k)).
2

%I #11 Feb 08 2022 23:16:26

%S 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,

%T 8,17,36,78,23,11,12,12,16,17,36,78,170,47,23,22,24,24,34,36,78,170,

%U 375,106,47,46,44,48,51,72,78,170,375,833

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

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

%F 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.

%e First few rows of the triangle =

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 1, 1, 2, 4;

%e 2, 1, 2, 4, 8;

%e 3, 2, 2, 4, 8, 17;

%e 6, 3, 4, 4, 8, 17, 36;

%e 11, 6, 6, 8, 8, 17, 36, 78;

%e 23, 11, 12, 12, 16, 17, 36, 78, 170;

%e 47, 23, 22, 24, 24, 34, 36, 78, 170, 375;

%e 106, 47, 46, 44, 48, 51, 72, 78, 170, 375, 833;

%e 235, 106, 94, 92, 88, 102, 108, 156, 170, 375, 833, 1870;

%e ...

%e Row 5 = (3, 2, 2, 4, 8, 17) = termwise products of (3, 2, 1, 1, 1, 1) and (1, 1, 2, 4, 8, 17).

%t 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 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];

%t u[n_] := u[n] = If[n <= 0, 1, Sum[t[i] u[n - i - 1], {i, 0, n}]];

%t 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);

%t 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 T[n_, k_] := v[n - k] u[k - 1];

%t 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 *)

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

%K nonn,tabl

%O 0,6

%A _Gary W. Adamson_, Mar 08 2009