A133179 -id:A133179

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A182033

Expansion of c(q^2)^2 / (c(q) * c(q^6)) in powers of q where c() is a cubic AGM theta function.

+10
4

1, -1, 1, 1, 0, -1, 1, 0, 0, -1, 0, 2, -1, 0, -2, 0, 0, -1, 1, 0, 4, 2, 0, -4, 0, 0, -1, -2, 0, 8, -3, 0, -8, -1, 0, -2, 4, 0, 14, 4, 0, -14, 1, 0, -4, -4, 0, 24, -6, 0, -23, -1, 0, -6, 5, 0, 40, 8, 0, -38, 1, 0, -10, -8, 0, 63, -10, 0, -60, -2, 0, -16, 11

(list; graph; refs; listen; history; text; internal format)

OFFSET

-1,12

LINKS

G. C. Greubel, Table of n, a(n) for n = -1..1000

FORMULA

Expansion of eta(q) * eta(q^6)^7 / (eta(q^2)^2 * eta(q^3)^3 * eta(q^18)^3) in powers of q.

Euler transform of period 18 sequence [ -1, 1, 2, 1, -1, -3, -1, 1, 2, 1, -1, -3, -1, 1, 2, 1, -1, 0, ...].

a(3*n) = 0 unless n=0. a(3*n + 1) = a(6*n + 2) = A092848(n). a(3*n + 2) = A062242(n). a(6*n + 4) = a(12*n + 8) = - A164614(n). a(6*n + 5) = A132179(n).

Convolution inverse of A122830.

EXAMPLE

1/q - 1 + q + q^2 - q^4 + q^5 - q^8 + 2*q^10 - q^11 - 2*q^13 - q^16 + ...

MATHEMATICA

eta[x_] := x^(1/24)*QPochhammer[x]; A182033[n_] := SeriesCoefficient[ eta[q]*eta[q^6]^7/(eta[q^2]^2*eta[q^3]^3*eta[q^18]^3 ), {q, 0, n}]; Table[A182033[n], {n, -1, 50}] (* G. C. Greubel, Aug 18 2017 *)

PROG

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^18 + A)^3), n))}

CROSSREFS

Cf. A062242, A092848, A122830, A133179, A164614.

KEYWORD

sign

AUTHOR

Michael Somos, Apr 07 2012

STATUS

approved

A130047

Left half of Pascal's triangle (A034868) modulo 2.

+10
2

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Row sums yield: 1, 1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 8, 1, 2, 2, 4, 2, 4, 4, 8, ...(see A048896).

LINKS

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

FORMULA

T(n,k) = mod(binomial(n, k), 2), 0 <= k <= floor(n/2). - G. C. Greubel, Aug 12 2017

EXAMPLE

Triangle begins:

1, 0,

1, 1,

1, 0, 0,

1, 1, 0,

1, 0, 1, 0,

1, 1, 1, 1,

1, 0, 0, 0, 0,

1, 1, 0, 0, 0,

1, 0, 1, 0, 0, 0,

1, 1, 1, 1, 0, 0,

1, 0, 0, 0, 1, 0, 0,

1, 1, 0, 0, 1, 1, 0,

1, 0, 1, 0, 1, 0, 1, 0,

1, 1, 1, 1, 1, 1, 1, 1,

1, 0, 0, 0, 0, 0, 0, 0, 0,

...

Triangle (right aligned) begins:

1, 0,

1, 1,

1, 0, 0,

1, 1, 0,

1, 0, 1, 0,

1, 1, 1, 1,

1, 0, 0, 0, 0,

1, 1, 0, 0, 0,

1, 0, 1, 0, 0, 0,

1, 1, 1, 1, 0, 0,

1, 0, 0, 0, 1, 0, 0,

1, 1, 0, 0, 1, 1, 0,

1, 0, 1, 0, 1, 0, 1, 0,

1, 1, 1, 1, 1, 1, 1, 1,

1, 0, 0, 0, 0, 0, 0, 0, 0,

1, 1, 0, 0, 0, 0, 0, 0, 0,

...

MAPLE

# From N. J. A. Sloane, Mar 22 2015:

for n from 0 to 20 do

lprint(seq(binomial(n, k) mod 2, k=0..floor(n/2))); od:

# For row sums:

f:=n->add(binomial(n, k) mod 2, k=0..floor(n/2));

[seq(f(n), n=0..60)];

MATHEMATICA

Table[Mod[Binomial[n, k], 2], {n, 0, 10}, {k, 0, Floor[n/2]}] (* G. C. Greubel, Aug 12 2017 *)

CROSSREFS

Cf. A007318, A034868, A048896, A133179.

KEYWORD

nonn,tabf

AUTHOR

Philippe Deléham, Oct 10 2007

EXTENSIONS

Corrected by N. J. A. Sloane, Mar 22 2015 at the suggestion of Kevin Ryde

STATUS

approved

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