A126713 The triangle K referred to in A038200, read along rows.

1, -1, 1, 1, -3, 1, -1, 7, -4, 1, 1, -15, 10, -5, 1, -1, 31, -19, 15, -6, 1, 1, -63, 28, -35, 21, -7, 1, -1, 127, -28, 71, -56, 28, -8, 1, 1, -255, 1, -135, 126, -84, 36, -9, 1, -1, 511, 80, 255, -251, 210, -120, 45, -10, 1, 1, -1023, -242, -495, 451, -462, 330, -165, 55, -11, 1, -1, 2047, 485, 991, -726, 925, -792, 495, -220

Offset

0,5

Comments

This means the description of A038200 is slightly incorrect and ought be: "Row sums of triangle K(m,n), inverse to a triangle obtained from A020921 after eliminating the leftmost column."

Links

Table of n, a(n) for n=0..74.

Formula

G.f.: (1/(1+x))*Sum(x^(k-1)/((1+x)^k-y*x^k),k=1..infinity). - Vladeta Jovovic, Feb 26 2008

Example

If the leftmost column of the triangle in A020921 is deleted we get
1
1 1
2 3 1
2 5 4 1
4 10 10 5 1
2 11 19 15 6 1
6 21 35 35 21 7 1
4 22 52 69 56 28 8 1
6 33 83 126 126 84 36 9 1
The present triangle is the inverse of this, namely
1
-1 1
1 -3 1
-1 7 -4 1
1 -15 10 -5 1
-1 31 -19 15 -6 1
1 -63 28 -35 21 -7 1
-1 127 -28 71 -56 28 -8 1
with row sums 1,0,-1,3,-8,21,-54,134,-318,720 of A038200.

Maple

A020921 := proc(n, k) option remember; local divs; if n <= 0 then 1; elif k > n then 0; else divs := numtheory[divisors](n); add(numtheory[mobius](op(i, divs))*binomial(n/op(i, divs), k), i=1..nops(divs)); fi; end: A020921t := proc(n, k) option remember; A020921(n+1, k+1); end: TriLInv := proc(nmax) local a, row, col; a := array(0..nmax, 0..nmax); for row from 0 to nmax do for col from row+1 to nmax do a[row, col] := 0; od; od; for row from 0 to nmax do for col from row to 0 by -1 do if row <> col then a[row, col] := -add(a[row, c]*A020921t(c, col), c=col+1..row)/A020921t(col, col); else a[row, col] := (1-add(a[row, c]*A020921t(c, col), c=col+1..row))/A020921t(col, col); fi; od; od; RETURN(a); end: nmax := 12 : a := TriLInv(nmax) : for row from 0 to nmax do for col from 0 to row do printf("%d, ", a[row, col]); od; od:

Mathematica

f[n_] := (1/(1+x))*Sum[x^(k-1)/((1+x)^k-y*x^k), {k, 1, n+1}]; t[0, 0] = 1; t[n_, k_] := SeriesCoefficient[f[n], {x, 0, n}, {y, 0, k}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 13 2013, after Vladeta Jovovic *)

Crossrefs

Cf. A039912.

Sequence in context: A135288 A078026 A350635 * A140068 A179745 A121300

Adjacent sequences: A126710 A126711 A126712 * A126714 A126715 A126716

Keyword

sign,tabl

Author

R. J. Mathar, Feb 12 2007

Status

approved