A156792 Triangle read by rows, T(n,k) = (A156791(n-k+1) * (A006973 * 0^(n-k))).

1, 1, 1, 6, 1, 2, 7, 6, 2, 9, 78, 7, 12, 9, 24, 420, 78, 14, 54, 24, 130, 6872, 420, 156, 63, 144, 130, 720, 17253, 6872, 840, 702, 168, 780, 720, 8505, 326552, 17253, 13744, 3780, 1872, 910, 4320, 8505, 35840

Offset: 0

Author

Gary W. Adamson, Feb 15 2009

Keywords

• eigen
• nonn
• tabl

Comments

This is a convolution triangle: A006973 convolved with A156791.

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

Examples

			First few rows of the triangle:
       1,
       1,     1;
       6,     1,     2;
       7,     6,     2,    9;
      78,     7,    12,    9,   24;
     420,    78,    14    54,   24, 130;
    6872,   420,   156,   63,  144, 130,  720;
   17253,  6872,   840,  702,  168, 780,  720, 8505;
  326552, 17253, 13744, 3780, 1872, 910, 4320, 8505, 35840;
  ...
Example: Row 4 = (7, 6, 2, 9) = termwise products of (7, 6, 1, 1) and (1, 1, 2, 9).
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A006973, A156791.

Programs

Mathematica

A006973[n_]:= A006973[n]= If[n<4, Max[n-1, 0], (n-1)!*(1 + Sum[k*(-A006973[k]/k!)^(n/k), {k, Most[Divisors[n]]}])];
S[n_, x_]:= Sum[A006973[j]*x^j, {j, 0, n+2}];
A156791:= With[{p=100}, CoefficientList[Series[S[p,x]/(x + S[p,x]), {x,0,p}], x]]
A156792[n_, k_]:= A156791[[n-k+2]]*(Boole[k==0] + A006973[k+1]);
Table[A156792[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 11 2021 *)

Formula

Triangle read by rows, T(n,k) = M*Q =(A156791(n-k+1) * (A006973 * 0^(n-k)))

M = an infinite lower triangular matrix with A156791: (1, 1, 6, 7, 78, ...) in every column.

Q = an infinite lower triangular matrix with A006973 prefaced with a 1 as the main diagonal: (1, 1, 2, 9, 24, 130, 720, 8505, ...) and the rest zeros.

Sum_{k=0..n} T(n, k) = A006973(n+1).

T(n, 0) = T(n+1, 1) = A156791(n).

T(n, k) = A156791(n-k+2)*([k=0] + A006973(k+1)). - G. C. Greubel, Jun 11 2021

Extensions

Typo in last line of triangle corrected by Olivier Gérard, Aug 11 2016