A381291 Number of subsets of 8 integers between 1 and n such that their sum is 0 modulo n.

1, 5, 15, 43, 99, 217, 429, 809, 1430, 2438, 3978, 6310, 9690, 14550, 21318, 30666, 43263, 60115, 82225, 111041, 148005, 195143, 254475, 328755, 420732, 534076, 672452, 840652, 1043460, 1287036, 1577532, 1922740, 2330445, 2810385, 3372291, 4028183, 4790071

Offset

9,2

Comments

For an integer s multiple of 8, this is also the number of subsets of 8 integers between 1 and n such that their sum is s modulo n.

Links

Table of n, a(n) for n=9..45.

David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025.

Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,11,-8,0,8,-10,0,8,0,-10,8,0,-8,11,-4,-4,4,-1).

Formula

G.f.: x^9*(1 + x - x^2 + 7*x^3 - 4*x^4 + 6*x^5 + 4*x^6 - 4*x^7 + 3*x^8 + 5*x^9 - 3*x^10 + x^11)/((1 - x)^4*(1 - x^2)^2*(1 - x^4)*(1 - x^8)).

a(n) = n^7/40320 - n^6/1440 + 23/2880*n^5 - 7/144*n^4 + O(n^3). - Charles R Greathouse IV, May 31 2026

Example

For n=10, there are a(10)=5 order 8 subsets of Z/10Z with sum equal to 0 mod 10.

Prog

(PARI) a(n)=(n^7-28*n^6+322*n^5-1960*n^4+(6874-n%2*105)*n^3+(-14392+n%2*1260)*n^2+42031616354777654719521424\1685^(n%8)%1685*12*n+5040*[-8, -1, -2, -1, -4, -1, -2, -1][n%8+1])/40320 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A381290, A381289, A011796.

Sequence in context: A381350 A111295 A200760 * A032193 A178965 A005665

Adjacent sequences: A381288 A381289 A381290 * A381292 A381293 A381294

Keyword

nonn,easy

Author

Xavier Roulleau and David Broadhurst, Feb 19 2025

Status

approved