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A173238 Triangle by columns, A000041 in every column shifted down twice for columns >0. 7
1, 1, 2, 1, 3, 1, 5, 2, 1, 7, 3, 1, 11, 5, 2, 1, 15, 7, 3, 1, 22, 11, 5, 2, 130, 15, 7, 3, 1, 42, 22, 11, 5, 2, 1, 56, 30, 15, 7, 3, 1, 77, 42, 22, 11, 5, 2, 1, 101, 56, 30, 15, 7, 3, 1, 135, 77, 42, 22, 11, 5, 2, 1, 176, 101, 56, 30, 15, 7, 3, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row sums = A024786 starting with A024786(2): (1, 1, 3, 4, 8, 11, 19, 26,...) = number of 2's in all partitions of n.

A173238 as an infinite lower triangular matrix * [1, 2, 3,...] = A103650.

Let the triangle = M. Then Lim_{n->inf} M^n = (1, 1, 3, 4, 10, 13, 26,...), the left-shifted vector considered as a sequence = A092119, the Euler transform of the ruler sequence, A001511.

Given P(x) = polcoeff A000041 = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...), then P(x) = A(x)/A(x^2), where A(x) = polcoeff A092119: (1 + x + 3x^2 + 4x^3 + ...).

Conjectures: Given the infinite set of triangles with A000041 in every column shifted down 0, 1, 2,... n times, row sums of n-th triangle (where A173238 = 2nd in the set) = the numbers of n's in all partitions of n. E.g., row sums = of A173238 = A024786, the numbers of 2's in all partitions of n.

Similarly, row sums of triangle A173239 with columns >0 shifted down thrice = numbers of 3's in all partitions of n, and so on. Refer to comments in A000041 regarding the numbers of 1's in partitions of n.

...

Second conjecture: Given the infinite set of analogous triangles with columns shifted down 2, 3, 4,...k times, we let such triangles = T(k) and perform Lim_{n->inf} T^n(k), obtaining the left-shifted vectors considered as sequences. The conjecture states that the infinite set of such left-shifted vectors = the Euler transform of the infinite set of Ruler functions starting with the ruler function for k=2 = A001511: (1, 2, 1, 3, 1, 2, 1,...)

To obtain the k-th ruler functions, begin with the natural numbers, 1,...2,...3,...4,...5,...6,...7,...8,...9,...; then for k = 2 we get A001511: 1,...2,...1,...3,...1,...2,...1,...4,...1,...; by finding the highest exponent of k dividing n, then adding 1. Similarly, for k = 2, we obtain A051064: 1,...1,...2,...1,...1,...2,...1,...1,...3,...

Next, we obtain the Euler transforms of the ruler functions (e.g., Euler transform of A001511 = A092119: (1, 1, 3, 4, 10, 13, 26,...), noting that A092119 is the Lim_{n->} A173238^n, the left-shifted vector.

...

Third conjecture: Let P(x) = polcoeff A000041 = (1 + x + 2x^2 + 3x^3 + ...), and A(k)(x) = the Euler transform of k-th ruler sequence, (k=2,3,...) Then P(x) = A(k)(x) / A(k)(x^k).

Examples: for k=2, A(x) = A092119: (1, 1, 3, 4, 10, 13,...), then P(x) = (1 + x + 2x^2 + 3x^3 + ...) = (1 + x + 3x^2 + 4x^3 + ...) / (1 + x^2 + 3x^4 + 4x^6 + 10x^8 + ...). For k=3 relating to triangle A173238, the left-shifted vector = the Euler transform of A051064 = A(x) for k=3, then P(x) = A(x) / A(x^3).

The conjecture extends the analogous conclusions to all k.

From Gary W. Adamson, Feb 25 2010: (Start)

Proof of second conjecture received from Helmut Prodinger 02/28/10 with a summary by R. J. Mathar:

Consider product_{n>=0} z^(t^n)/(1-z^(t^n)) = sum_{k>=1} (1+v_t(k))z^k where v_t(n) is the number of trailing zeros in the t-ary expansion of n,and its Euler transform A(z) = product_{k >= 1} 1/(1-z^k)^{1+v_t(k)}, then A(z)/A(z^t) = product_{k >= 1} 1/(1-z^k) is the partition generating function.

Here is the proof:A(z)/A(z^t) = product_{k>=1} (1-z^(tk))^{1+v_t(k)}/(1-z^k)^{1+v_t(k)}

= \product_{k>=1} (1-z^(tk))^{v_t(tk)}/(1-z^k)^{1+v_t(k)}

= \product_{k>=1} (1-z^k)^{v_t(k)}/(1-z^k)^{1+v_t(k)} (*)

= \product_{k>=1} 1/(1-z^k)

as desired. Notice that for (*), that v_t(n)=0 if n is not divisible by t. [Helmut Prodinger, hproding(AT)sun.ac.za, Feb 28 2010] (End)

LINKS

Table of n, a(n) for n=0..70.

FORMULA

Triangle by columns, A000041 in every column shifted down twice for columns >0.

EXAMPLE

First few rows of the triangle =

1;

1;

2, 1;

3, 1;

5, 2, 1;

7, 3, 1;

11, 5, 2, 1;

15, 7, 3, 1;

22, 11, 5, 2, 1;

30, 15, 7, 3, 1;

42, 22, 11, 5, 2, 1;

56, 30, 15, 7, 3, 1;

77, 42, 22, 11, 5, 2, 1;

101, 56, 30, 15, 7, 3, 1;

135, 77, 42, 22, 11, 5, 2, 1;

176, 101, 56, 30, 15, 7, 3, 1;

...

CROSSREFS

Cf. A000041, A001511, A092119, A173239.

Sequence in context: A147000 A147486 A168018 * A173284 A278136 A085053

Adjacent sequences:  A173235 A173236 A173237 * A173239 A173240 A173241

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Feb 13 2010

STATUS

approved

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Last modified March 26 10:42 EDT 2017. Contains 284111 sequences.