login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A294746 Number A(n,k) of compositions (ordered partitions) of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals. 21
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 10, 13, 1, 1, 1, 35, 217, 75, 1, 1, 1, 126, 4245, 8317, 525, 1, 1, 1, 462, 90376, 1239823, 487630, 4347, 1, 1, 1, 1716, 2019836, 216456376, 709097481, 40647178, 41245, 1, 1, 1, 6435, 46570140, 41175714454, 1303699790001, 701954099115, 4561368175, 441675, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
COMMENTS
Row r >= 2, is asymptotic to r^(r*n + 3/2) / (2*Pi*n)^((r-1)/2). - Vaclav Kotesovec, Sep 20 2019
LINKS
FORMULA
A(n,k) = [x^((k+1)^n)] (Sum_{j=0..k*n+1} x^((k+1)^j))^(k*n+1) for k>0, A(n,0) = 1.
EXAMPLE
A(3,1) = 13: [1/4,1/4,1/4,1/4], [1/2,1/4,1/8,1/8], [1/2,1/8,1/4,1/8], [1/2,1/8,1/8,1/4], [1/4,1/2,1/8,1/8], [1/4,1/8,1/2,1/8], [1/4,1/8,1/8,1/2], [1/8,1/2,1/4,1/8], [1/8,1/2,1/8,1/4], [1/8,1/4,1/2,1/8], [1/8,1/4,1/8,1/2], [1/8,1/8,1/2,1/4], [1/8,1/8,1/4,1/2].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 3, 10, 35, 126, 462, ...
1, 13, 217, 4245, 90376, 2019836, ...
1, 75, 8317, 1239823, 216456376, 41175714454, ...
1, 525, 487630, 709097481, 1303699790001, 2713420774885145, ...
MAPLE
b:= proc(n, r, p, k) option remember;
`if`(n<r, 0, `if`(r=0, `if`(n=0, p!, 0), add(
b(n-j, k*(r-j), p+j, k)/j!, j=0..min(n, r))))
end:
A:= (n, k)-> `if`(k=0, 1, b(k*n+1, 1, 0, k+1)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
A[n_, k_] := If[k == 0, 1, b[k*n + 1, 1, 0, k + 1]];
Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)
CROSSREFS
Main diagonal gives: A294747.
Cf. A294775.
Sequence in context: A263383 A185620 A096066 * A326180 A064085 A256692
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Nov 07 2017
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 23 20:31 EST 2024. Contains 370288 sequences. (Running on oeis4.)