cp's OEIS Frontend

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Also numbers k such that the k-th composition in standard order has reverse-alternating sum != 0.

a(n) is twice the area of the trapezoid created by the four points (2^n,2^(n+1)), (2^(n+1), 2^n), (2^(n+1), 2^(n+2)), (2^(n+2), 2^(n+1)). - J. M. Bergot, May 23 2014

These are squares that can be expressed as sum of exactly two distinct powers of two. For instance, a(4) = 9*4^4 = 2304 = 2^11 + 2^8 . It is conjectured that these are the only squares with this characteristic (tested on squares up to (10^7)^2). - Andres Cicuttin, Apr 23 2016

Conjecture is true. It is equivalent to prove that the Diophantine equation m^2 = 2^k*(1+2^h), where h>0, has solutions only when h=3. Dividing by 2^k we must obtain an odd square on the left, since 1+2^h is odd, so we can write (2*r+1)^2 = 1+2^h. Expanding, we have 4*r*(r+1) = 2^h, from which it follows that r must be equal to 1 and thus h=3, since r and r+1 must be powers of 2. - Giovanni Resta, Jul 27 2017

Or, constant term in expansion of Product_{k=1..n} (x^k + 1/x^k)^2. - N. J. A. Sloane, Jul 09 2008

Or, maximal coefficient of the polynomial (1+x)^2 * (1+x^2)^2 *...* (1+x^n)^2.

a(n) = A000302(n) - A181765(n).

From Gus Wiseman, Apr 18 2023: (Start)

Also the number of subsets of {1..2n} that are empty or have mean n. The a(0) = 1 through a(3) = 10 subsets are:

{} {} {} {}

{1} {2} {3}

{1,3} {1,5}

{1,2,3} {2,4}

{1,2,6}

{1,3,5}

{2,3,4}

{1,2,3,6}

{1,2,4,5}

{1,2,3,4,5}

Also the number of subsets of {-n..n} with no 0's but with sum 0. The a(0) = 1 through a(3) = 10 subsets are:

{} {} {} {}

{-1,1} {-1,1} {-1,1}

{-2,2} {-2,2}

{-2,-1,1,2} {-3,3}

{-3,1,2}

{-2,-1,3}

{-2,-1,1,2}

{-3,-1,1,3}

{-3,-2,2,3}

{-3,-2,-1,1,2,3}

(End)

Number of palindromes of length n using a maximum of four different symbols.

Number of achiral rows of n colors using up to four colors. - Robert A. Russell, Nov 09 2018

Interleaving of A000302 and 4*A000302.

Unsigned version of A141125.

Binomial transform is A164907. Second binomial transform is A164908. Third binomial transform is A057651. Fourth binomial transform is A016129.

The row polynomials Pe(n, x) := Sum_{m=0..n} a(n, m)*x^m appear as numerators of the generating functions for the even numbered column sequences of array A034870.

Elements have the same parity as those of Pascal's triangle.

All zeros of polynomial Pe(n, x) are negative. They are -tan^2(Pi/2*n+1), -tan^2(2*Pi/2*n+1), ..., -tan^2(n*Pi/2*n+1). Moreover, for m >= 1, Pe(m, -x^2) is the characteristic polynomial of the linear difference equation with constant coefficients for differences between multiples of 2*m+1 with even and odd digit sum in base 2*m in the interval [0,(2*m)^n). - Vladimir Shevelev and Peter J. C. Moses, May 22 2012

Row reverse of A103327. - Peter Bala, Jul 29 2013

The row polynomial Pe(d, x), multiplied by (2*d)!/d! = A001813(d), gives the numerator polynomial of the o.g.f. of the sequence of the diagonal d, for d >= 0, of the Sheffer triangle Lah[4,1] given in A048854. - Wolfdieter Lang, Oct 12 2017

Partial sums are A097074.

Pairwise sums are {1, 1, 4, 16, 32, ...} or 2^n -Sum_{k=0..n} binomial(n,k)*(-1)^(n+k)*k.

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:

a\b 1.......2.......3.......4.......5.......6

-1 A144431

0 A007318 A038208 A038221

1 A008292 A256890 A257180 A257606 A257607

2 A060187 A257609 A257611 A257613 A257615

3 A142458 A257610 A257620 A257622 A257624 A257626

4 A142459 A257612 A257621

5 A142460 A257614 A257623

6 A142461 A257616 A257625

7 A142462 A257617 A257627

8 A167884 A257618

9 A257608 A257619

The row sums of these, and similarly constructed number triangles, are shown in the following table:

a\b 1.......2.......3.......4.......5.......6.......7.......8.......9

0 A000079 A000302 A000400

1 A000142 A001715 A001725 A049388 A049198

2 A000165 A002866 A002866 A051580 A051582

3 A008544 A051578 A037559 A051605 A051607 A051609

4 A001813 A047053 A000407 A034177 A051618 A051620 A051622

5 A047055 A008546 A008548 A034300 A034325 A051688 A051690

6 A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103

7 A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105

8 A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115

9 A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023

10 A051262 A035265 A035273 A035277

11 A254322

12 A145448

The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).

a\b 0 1 2 3

-2 A130595/1

-1

0

1 A110555/-1 A120434/-1 A144697/1 A144699/2

With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash).

If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).

In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019