A208654 -id:A208654 - cp's OEIS frontend

A208650 Number of constant paths through the subset array of {1,2,...,n}; see Comments.

1, 2, 6, 36, 480, 15000, 1134000, 211768200, 99131719680, 117595223746560, 356467003200000000, 2779532232516963000000, 56049508602150185041920000, 2935889842347365340037522521600

Offset: 1

Clark Kimberling, Mar 01 2012

nonn

Let I(n)={1,2,...,n}. Arrange the subsets of I(n) in an array S(n) of n rows, where row k consists of all the numbers in all the k-element subsets, including repetitions. Each i in I(n) occurs C(n-1,k-1) times in row k of S(n); index these occurrences as
...
(k,1,1),(k,1,2),...,(k,1,r),(k,2,1),...,(k,2,r),...,(k,n,1),...,(k,n,r),
...
where r=C(n-1,k-1).
Definitions:
(1) A path through I(n) is an n-tuple of triples, ((1,i(1),j(1)), (2,i(2),j(2)), ..., (n,i(n),j(n))), formed from the above indexing of the numbers in S(n).
(2) The trace of such a path p is the n-tuple (i(1),i(2),...,i(n)).
(3) The range of p is the set {i(1),i(2),...,i(n)}.
(4) Path p has property P if its trace or range has property P.
...
Guide to sequences which count paths according to selected properties:
property................................sequence
range = {1}.............................A001142(n-1)
constant (range just one element).......A208650
range = {1,2,...,n}.....................A208651
palindromic.............................A208654
palindromic with i(1)=1.................A208655

			Taking n=3:
row 1:  {1},{2},{3} ---------> 1,2,3
row 2:  {1,2},{1,3},{2,3} ---> 1,1,2,2,3,3
row 3:  {1,2,3} -------------> 1,2,3
3 ways to choose a number from row 1,
2 ways to choose same number from row 2,
1 way to choose same number from row 3.
Total:  a(3) = 1*2*3 = 6 paths.

Cf. A208651.

p[n_]:=Product[Binomial[n-1,k],{k,1,n-1}]
Table[p[n],{n,1,20}]    (* A001142(n-1) *)
Table[p[n]*n,{n,1,20}]  (* A208650 *)
Table[p[n]*n!,{n,1,20}] (* A208651 *)

a(n) = n*Product_{k=1..n-1} binomial(n-1,k). - Jason Yuen, Feb 18 2025

A208652 Product{i*C(n,i) : 1<=i<=floor[(n+1)/2]}.

Original entry on oeis.org

1, 2, 18, 48, 3000, 10800, 4321800, 21073920, 51849745920, 342921600000, 5062554438480000, 45444644517888000, 3960533401513405516800, 48272757747179554068480, 24547182197052103038450000000

Offset: 1

Clark Kimberling, Mar 02 2012

nonn

Used, along with A208653, to count palindromic paths in A208654.

Cf. A208653, A208654.

m[n_]:=Floor[(n+1)/2];z=21;
g[n_]:=Product[i*Binomial[n,i],{i,1,m[n]}]
Table[g[n],{n,1,z}] (* A208652 *)
Table[Gamma[1 + n]^(n/2)/BarnesG[1 + n]*((1 + (-1)^n)/2 + (1 - (-1)^n) * Sqrt[Gamma[1 + n]]/(2*Gamma[(1 + n)/2])), {n, 1, 15}] (* Vaclav Kotesovec, Apr 10 2024 *)

A208653 a(n) = Product_{i=floor((n + 1)/2)..n-1} binomial(n-1, i).

Original entry on oeis.org

1, 1, 1, 3, 4, 50, 90, 5145, 12544, 3429216, 11340000, 15219319500, 68309049600, 457937132487120, 2790771598030416, 94609025993497640625, 783056974947287040000, 135476575389769051389952000, 1523136299736565293430210560, 1354434926051634531310373234715648

Offset: 1

Clark Kimberling, Mar 02 2012

nonn

Used with A208652 to count palindromic paths for A208654.

Cf. A208652, A208654.

m[n_]:=Floor[(n+1)/2]; z=21;
h[n_]:=Product[Binomial[n-1,i],{i,m[n],n-1}]
Table[h[n],{n,1,z}] (* A208653 *)

Definition corrected by Georg Fischer, Feb 01 2022

A208655 Number of palindromic paths starting with 1 through the subset array of {1,2,...,n}; see Comments.

Original entry on oeis.org

1, 1, 6, 36, 2400, 90000, 55566000, 13553164800, 72267023646720, 117595223746560000, 5219033393851200000000, 57636380373471744768000000, 20810790197418148654769602560000, 1578992018570629416640340512656998400

Offset: 1

Clark Kimberling, Mar 02 2012

nonn

A palindromic path through the subset array of {1,2,...,n} is essentially a palindrome using numbers i from {1,2,...n}, where the number of times i can be used in position k equals the multiplicity of i in the multiset of numbers in the k-element subsets of {1,2,...,n}. See A208650 for a discussion and guide to related sequences.

			For n=4, write
row 1:  1,2,3,4
row 2:  1,2; 1,3; 1,4; 2,3; 2,4; 3;4
row 3:  1,2,3; 1,2,4; 1,3,4; 2,3,4
row 4:  1,2,3,4
To form a palindromic path of length 4 and starting with 1, there is 1 way to choose 1st term from row 1, then 12 ways to choose 2nd term from row 2, then 3 ways to choose 3rd term, then 1 way to finish.  Thus, a(4)=1*12*3*1=36.

Cf. A208650, A208654.

m[n_] := Floor[(n + 1)/2]; z = 21;
g[n_] := Product[i*Binomial[n, i], {i, 1, m[n]}]
h[n_] := Product[Binomial[n - 1, i], {i, m[n], n - 1}]
Table[g[n], {n, 1, z}]   (* A208652 *)
Table[h[n], {n, 1, z}]   (* A208653 *)
Table[g[n] h[n], {n, 1, 2 z/3}]   (* A208654 *)
Table[g[n] h[n]/n, {n, 1, 2 z/3}] (* A208655 *)

cp's OEIS Frontend

A208650 Number of constant paths through the subset array of {1,2,...,n}; see Comments.

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A208652 Product{i*C(n,i) : 1<=i<=floor[(n+1)/2]}.

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A208653 a(n) = Product_{i=floor((n + 1)/2)..n-1} binomial(n-1, i).

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A208655 Number of palindromic paths starting with 1 through the subset array of {1,2,...,n}; see Comments.

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