A380886 Triangle T(n,k), 1<=k<=n: column k are the coefficients of the INVERT transform of Sum_{i=1..k} i*x^i.

1, 1, 3, 1, 5, 8, 1, 11, 17, 21, 1, 21, 42, 50, 55, 1, 43, 100, 128, 138, 144, 1, 85, 235, 323, 358, 370, 377, 1, 171, 561, 813, 923, 965, 979, 987, 1, 341, 1331, 2043, 2378, 2510, 2559, 2575, 2584, 1, 683, 3158, 5150, 6125, 6527, 6681, 6737, 6755, 6765, 1, 1365, 7503, 12967, 15772, 16972, 17441, 17617, 17680, 17700, 17711

Offset: 1

Author

R. J. Mathar, Feb 07 2025

Keywords

• nonn
• tabl
• easy

Examples

			The full array starts
  1    1    1    1    1    1    1    1    1    1
  1    3    3    3    3    3    3    3    3    3
  1    5    8    8    8    8    8    8    8    8
  1   11   17   21   21   21   21   21   21   21
  1   21   42   50   55   55   55   55   55   55
  1   43  100  128  138  144  144  144  144  144
  1   85  235  323  358  370  377  377  377  377
  1  171  561  813  923  965  979  987  987  987
  1  341 1331 2043 2378 2510 2559 2575 2584 2584
  1  683 3158 5150 6125 6527 6681 6737 6755 6765
but the non-interesting upper right triangular part is not put into the sequence.
		

Crossrefs

Cf. A001045 (column k=2), A101822 (column k=3), A322059 (column k=4?), A001906 (diagonal), A054452 (subdiagonal).

Programs

Maple

A380886 := proc(n,k)
    local g,x ;
    g := 1/(1-add(i*x^i,i=1..k)) ;
    coeftayl(g,x=0,n) ;
end proc:
seq(seq( A380886(n,k),k=1..n),n=1..12) ;

Formula

T(n,k) = [x^n] 1/(1-x^1-2*x^2-3*x^3-4*x^4-...-k*x^k) .