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A104509 Matrix inverse of triangle A104505, which is the right-hand side of triangle A084610 of coefficients in (1+x-x^2)^n. 6
1, 1, -1, 3, -2, 1, 4, -6, 3, -1, 7, -12, 10, -4, 1, 11, -25, 25, -15, 5, -1, 18, -48, 60, -44, 21, -6, 1, 29, -91, 133, -119, 70, -28, 7, -1, 47, -168, 284, -296, 210, -104, 36, -8, 1, 76, -306, 585, -699, 576, -342, 147, -45, 9, -1, 123, -550, 1175, -1580, 1485, -1022, 525, -200, 55, -10, 1, 199, -979, 2310, -3454, 3641 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Leftmost column is A000204 (Lucas numbers). Other columns include: A045925, A067988. Row sums are: {1,0,2,0,2,0,2,...}. Absolute row sums form: A099425. Antidiagonal sums are: {1,1,2,2,2,2,2,...}. Absolute antidiagonal sums are: A084214.

Riordan array ( (1 + x^2/(1 - x - x^2), -x/(1 - x - x^2) ) belonging to the hitting time subgroup of the Riordan group (see Peart and Woan). - Peter Bala, Jun 29 2015

LINKS

Robert Israel, Table of n, a(n) for n = 0..10152 (rows 0 to 141, flattened).

P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

FORMULA

G.f.: A(x, y) = (1 + x^2)/(1-x-x^2 + x*y).

G.f. for column k: g_k(x) = -(x^2+1)*x^k/(x^2+x-1)^(k+1). - Robert Israel, Jun 30 2015

EXAMPLE

Rows begin:

1;

1,-1;

3,-2,1;

4,-6,3,-1;

7,-12,10,-4,1;

11,-25,25,-15,5,-1;

18,-48,60,-44,21,-6,1;

29,-91,133,-119,70,-28,7,-1;

47,-168,284,-296,210,-104,36,-8,1;

76,-306,585,-699,576,-342,147,-45,9,-1; ...

MAPLE

S:= series((1 + x^2)/(1-x-x^2 + x*y), x, 20):

for n from 0 to 19 do

R[n]:= coeff(S, x, n)

od:

seq(seq(coeff(R[n], y, j), j=0..n), n=0..19); # Robert Israel, Jun 30 2015

MATHEMATICA

nmax = 11;

T[n_, k_] := Coefficient[(1 + x - x^2)^n, x, n + k];

M = Table[T[n, k], {n, 0, nmax}, {k, 0, nmax}] // Inverse;

Table[M[[n+1, k+1]], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Mar 27 2019 *)

PROG

(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff((1 + X^2)/(1-X-X^2 + X*Y), n, x), k, y)}

(PARI) tabl(nn) = {my(m = matrix(nn, nn, n, k, n--; k--; if((n<k)||(k<0), 0, polcoeff((1+x-x^2)^n, n+k, x)))^(-1)); for (n=1, nn, for (k=1, n, print1(m[n, k], ", "); ); print(); ); } \\ Michel Marcus, Jun 30 2015

CROSSREFS

Cf. A104505, A000204.

Sequence in context: A222220 A271830 A193815 * A271513 A306801 A117212

Adjacent sequences:  A104506 A104507 A104508 * A104510 A104511 A104512

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Mar 11 2005

STATUS

approved

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Last modified July 28 00:54 EDT 2021. Contains 346316 sequences. (Running on oeis4.)