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A160096 Partial sums of A010815 starting with offset 1, and signed (+ + - - + + ...). 3
1, 2, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

INVERT transform of the sequence = A137682: (1, 3, 7, 17, 40, 96, 228,...).

From Mats Granvik, Jan 01 2015: (Start)

(1) The natural numbers are the row sums of the infinite lower triangular matrix "t" starting:

1,0,0,0,0,0,0,...

1,1,0,0,0,0,0,...

1,1,1,0,0,0,0,...

1,1,1,1,0,0,0,...

1,1,1,1,1,0,0,...

1,1,1,1,1,1,0,...

1,1,1,1,1,1,1,...

.................

which satisfies the recurrence:

t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, n - 1}] - Sum[t[n - i, k], {i, 1, n - 1}], 0];

(2) This sequence a(n), in turn, is the row sums of the infinite lower triangular matrix "t" starting:

1,0,0,0,0,0,0,...

1,1,0,0,0,0,0,...

1,0,1,0,0,0,0,...

1,0,0,1,0,0,0,...

1,0,-1,0,1,0,0,...

1,0,0,-1,0,1,0,...

1,0,0,-1,-1,0,1,...

...................

which satisfies the recurrence:

t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}] - Sum[t[n - i, k], {i, 1, n - 1}], 0];

(3) The partition numbers are the row sums of the infinite lower triangular matrix "t" starting:

1,0,0,0,0,0,0,...

1,1,0,0,0,0,0,...

1,1,1,0,0,0,0,...

1,2,1,1,0,0,0,...

1,2,2,1,1,0,0,...

1,3,3,2,1,1,0,...

1,3,4,3,2,1,1,...

.................

which satisfies the recurrence:

t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, n - 1}] - Sum[t[n - i, k], {i, 1, k - 1}], 0];

(4) The number of divisors of "n" is

the row sums of the infinite lower triangular matrix "t" starting:

1,0,0,0,0,0,0,...

1,1,0,0,0,0,0,...

1,0,1,0,0,0,0,...

1,1,0,1,0,0,0,...

1,0,0,0,1,0,0,...

1,1,1,0,0,1,0,...

1,0,0,0,0,0,1,...

.................

which satisfies the recurrence:

t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}] - Sum[t[n - i, k], {i, 1, k - 1}], 0].

In the four cases of recurrences only the summation index within the sums change, from (1) "n-1" and "n-1" to (2) "k-1" and "n-1" to (3) "n-1" and "k-1" to (4) "k-1" and "k-1".

(End)

LINKS

Table of n, a(n) for n=1..90.

FORMULA

Partial sums of Euler's q series (signed), starting from offset 1 = (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, 1, ...).

G.f.: (1 - f(-x)) / (1 - x) where f(-x) is the g.f. of A010815. - Michael Somos, Jan 02 2015

EXAMPLE

The series begins (1, 2, 2, 2, 1, 1, 0, ...) since the signed q-series = (1, 1, 0, 0, -1, 0, ...).

G.f. = x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6 + x^12 + x^13 + x^14 + ...

MATHEMATICA

(*A160096 as row sums of recursively defined table*) Clear[t]; nn = 90; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}] - Sum[t[n - i, k], {i, 1, n - 1}], 0]; PartialSumsOfEulerqSeries = Table[Sum[t[n, k], {k, 1, n}], {n, 1, nn}] (* Mats Granvik, Jan 01 2015 *)

a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ x]) / (1 - x), {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)

CoefficientList[Series[q*(1/(1 - q)^(2)*QHypergeometricPFQ[{q, q}, {q^2, q}, q, q^2]), {q, 0, 89}], q] (* Mats Granvik, Jan 09 2015 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( (1 - eta(x + x * O(x^n))) / (1 - x), n))}; /* Michael Somos, Jan 02 2015 */

CROSSREFS

Cf. A101815.

Cf. (1) A000027, (2) A160096, (3) A000041, (4) A000005.

Sequence in context: A058101 A132980 A106823 * A029446 A288160 A275332

Adjacent sequences:  A160093 A160094 A160095 * A160097 A160098 A160099

KEYWORD

nonn

AUTHOR

Gary W. Adamson, May 01 2009

EXTENSIONS

More terms from Mats Granvik, Jan 01 2015

STATUS

approved

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Last modified July 17 09:23 EDT 2019. Contains 325100 sequences. (Running on oeis4.)