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A347580 Triangle read by rows: T(n,k) is the number of chains of length k in the poset of all arithmetic progressions contained in {1,...,n} of length in the range [1..n-1], ordered by inclusion. 0
1, 1, 2, 1, 6, 6, 1, 12, 24, 12, 1, 21, 68, 72, 24, 1, 32, 144, 244, 180, 48, 1, 47, 283, 666, 764, 432, 96, 1, 64, 486, 1510, 2436, 2164, 1008, 192, 1, 85, 799, 3117, 6534, 8028, 5816, 2304, 384, 1, 109, 1232, 5860, 15368, 24524, 24516, 15040, 5184, 768 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Let L_n be the lattice of all arithmetic progressions contained in {1,...,n}, including the empty progression and the whole interval. T(n,k) is the number of chains of length k+2 in L_n that contain both the maximal and minimal element.

LINKS

Table of n, a(n) for n=1..55.

M. K. Goh, J. Hamdan, and J. Saks, The lattice of arithmetic progressions, arXiv:2106.05949 [math.CO], 2021. See Table 2 p. 7.

FORMULA

Let f(n,k) = n, if k=1; A338993(n,k)/2, if 2<=k<=n. Then T(n,k) = 1, if k=1; Sum_{i=1..n-1} f(n,k)*T(i,k-1), if 2<=k<=n; 0, if k>n.

Sum_{k=1..n} (-1)^k*T(n,k) = A008683(n-1), for n>=2.

EXAMPLE

Triangle begins:

  n/k 1   2    3     4     5      6      7      8      9    10    11   12

   1  1

   2  1   2

   3  1   6    6

   4  1  12   24    12

   5  1  21   68    72    24

   6  1  32  144   244   180     48

   7  1  47  283   666   764    432     96

   8  1  64  486  1510  2436   2164   1008    192

   9  1  85  799  3117  6534   8028   5816   2304    384

  10  1 109 1232  5860 15368  24524  24516  15040   5184   768

  11  1 137 1838 10418 33049  65402  84284  70992  37760 11520  1536

  12  1 167 2611 17420 65706 157010 250332 270996 197280 92608 25344 3072

MATHEMATICA

t[n_, k_] := If[k == 1, n, Sum[2(n-(k-1) r), {r, 1, Quotient[n-1, k-1]}]];

f[n_, k_] := If[k == 1, n, t[n, k]/2];

T[n_, k_] := T[n, k] = If[k == 1, 1, Sum[f[n, i] T[i, k-1], {i, 1, n-1}]];

Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 13 2021, from PARI code *)

PROG

(PARI) t(n, k) = if (k==1, n, sum(r=1, (n-1)\(k-1), 2*(n-(k-1)*r))); \\ A338993

f(n, k) = if (k==1, n, t(n, k)/2);

T(n, k) = if (k==1, 1, sum(i=1, n-1, f(n, i)*T(i, k-1))); \\ Michel Marcus, Sep 11 2021

CROSSREFS

Cf. A008683, A338993.

Sequence in context: A229565 A259477 A208919 * A259569 A046651 A063007

Adjacent sequences:  A347577 A347578 A347579 * A347581 A347582 A347583

KEYWORD

nonn,tabl

AUTHOR

Marcel K. Goh, Sep 07 2021

STATUS

approved

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Last modified October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)