A116406 - OEIS

A116406

Expansion of ((1 + x - 2x^2) + (1+x)*sqrt(1-4x^2))/(2(1-4x^2)).

1, 1, 2, 3, 7, 11, 26, 42, 99, 163, 382, 638, 1486, 2510, 5812, 9908, 22819, 39203, 89846, 155382, 354522, 616666, 1401292, 2449868, 5546382, 9740686, 21977516, 38754732, 87167164, 154276028, 345994216, 614429672, 1374282019, 2448023843

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

OFFSET

0,3

COMMENTS

Interleaving of A114121 and A032443. Row sums of A116405. Binomial transform is A116409.

Appears to be the number of n-digit binary numbers not having more zeros than ones; equivalently, the number of unrestricted Dyck paths of length n not going below the axis. - Ralf Stephan, Mar 25 2008

From Gus Wiseman, Jun 20 2021: (Start)

Also the number compositions of n with alternating sum >= 0, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. The a(0) = 1 through a(5) = 11 compositions are:

() (1) (2) (3) (4) (5)

(11) (21) (22) (32)

(111) (31) (41)

(112) (113)

(121) (122)

(211) (212)

(1111) (221)

(311)

(1121)

(2111)

(11111)

(End)

From J. Stauduhar, Jan 14 2022: (Start)

Also, for n >= 2, first differences of partial row sums of Pascal's triangle. The first ceiling(n/2)+1 elements of rows n=0 to n=4 in Pascal's triangle are:

1 1

1 2

1 3 3

1 4 6

...

The cumulative sums of these partial rows form the sequence 1,3,6,13,24,..., and its first differences are a(2),a(3),a(4),... in this sequence.

(End)

LINKS

Table of n, a(n) for n=0..33.

FORMULA

a(n) = A114121(n/2)*(1+(-1)^n)/2 + A032443((n-1)/2)*(1-(-1)^n)/2.

a(n) = Sum_{k=0..floor(n/2)} binomial(n-1,k). - Paul Barry, Oct 06 2007

Conjecture: n*(n-3)*a(n) +2*(-n^2+4*n-2)*a(n-1) -4*(n-2)^2*a(n-2) +8*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 28 2014

a(n) ~ 2^(n-2) * (1 + (3+(-1)^n)/sqrt(2*Pi*n)). - Vaclav Kotesovec, May 30 2016

a(n) = 2^(n-1) - A294175(n). - Gus Wiseman, Jun 27 2021

MATHEMATICA

CoefficientList[Series[((1+x-2x^2)+(1+x)Sqrt[1-4x^2])/(2(1-4x^2)), {x, 0, 40}], x] (* Harvey P. Dale, Aug 16 2012 *)

ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}]; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], ats[#]>=0&]], {n, 0, 15}] (* Gus Wiseman, Jun 20 2021 *)

CROSSREFS

The alternating sum = 0 case is A001700 or A088218.

The alternating sum > 0 case appears to be A027306.

The bisections are A032443 (odd) and A114121 (even).

The alternating sum <= 0 version is A058622.

The alternating sum < 0 version is A294175.

The restriction to reversed partitions is A344607.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A124754 gives the alternating sum of standard compositions.

A344610 counts partitions by sum and positive reverse-alternating sum.

A344616 lists the alternating sums of partitions by Heinz number.