A318431 -id:A318431

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A304482

Number A(n,k) of n-element subsets of [k*n] whose elements sum to a multiple of n. Square array A(n,k) with n, k >= 0 read by antidiagonals.

+10
9

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 1, 0, 1, 4, 6, 8, 0, 0, 1, 5, 12, 30, 18, 1, 0, 1, 6, 20, 76, 126, 52, 0, 0, 1, 7, 30, 155, 460, 603, 152, 1, 0, 1, 8, 42, 276, 1220, 3104, 3084, 492, 0, 0, 1, 9, 56, 448, 2670, 10630, 22404, 16614, 1618, 1, 0, 1, 10, 72, 680, 5138, 28506, 98900, 169152, 91998, 5408, 0, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

When k=1 the only subset of [n] with n elements is [n] which sums to n(n+1)/2 and hence for n>0 and n even A(n,1) is zero and for n odd A(n,1) is one.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Marko Riedel et al., Number of n-element subsets divisible by n

FORMULA

A(n,k) = (-1)^n * (1/n) * Sum_{d|n} C(k*d,d)*(-1)^d*phi(n/d), boundary values A(0,0) = 1, A(n, 0) = 0, A(0, k) = 1.

EXAMPLE

Square array A(n,k) begins:

1, 1, 1, 1, 1, 1, 1, 1, ...

0, 1, 2, 3, 4, 5, 6, 7, ...

0, 0, 2, 6, 12, 20, 30, 42, ...

0, 1, 8, 30, 76, 155, 276, 448, ...

0, 0, 18, 126, 460, 1220, 2670, 5138, ...

0, 1, 52, 603, 3104, 10630, 28506, 64932, ...

0, 0, 152, 3084, 22404, 98900, 324516, 874104, ...

0, 1, 492, 16614, 169152, 960650, 3854052, 12271518, ...

MAPLE

with(numtheory):

A:= (n, k)-> `if`(n=0, 1, add(binomial(k*d, d)*(-1)^(n+d)*

phi(n/d), d in divisors(n))/n):

seq(seq(A(n, d-n), n=0..d), d=0..11);

MATHEMATICA

A[n_, k_] : = (-1)^n (1/n) Sum[Binomial[k d, d] (-1)^d EulerPhi[n/d], {d, Divisors[n]}]; A[0, 0] = 1; A[_, 0] = 0; A[0, _] = 1;

Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 23 2019 *)

PROG

(PARI) T(n, k)=if(n==0, 1, (-1)^n*sumdiv(n, d, binomial(k*d, d) * (-1)^d * eulerphi(n/d))/n)

for(n=0, 7, for(k=0, 7, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Aug 28 2018

CROSSREFS

Cf. A000007, A169888, A318431, A318432, A318433, A318557.

Main diagonal gives A318477.

KEYWORD

nonn,tabl

AUTHOR

Marko Riedel, Aug 28 2018

STATUS

approved

A169888

Number of n-member subsets of 1..2n whose elements sum to a multiple of n.

+10
8

1, 2, 2, 8, 18, 52, 152, 492, 1618, 5408, 18452, 64132, 225432, 800048, 2865228, 10341208, 37568338, 137270956, 504171584, 1860277044, 6892335668, 25631327688, 95640829924, 357975249028, 1343650267288, 5056424257552, 19073789328752, 72108867620204

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

This is twice A145855 (for n>0), which is the main entry for this problem.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1669

FORMULA

a(n) = A061865(2n,n). - Alois P. Heinz, Aug 28 2018

a(n) ~ 2^(2*n) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 28 2023

MAPLE

with(combinat): t0:=[]; for n from 1 to 8 do ans:=0; t1:=choose(2*n, n); for i in t1 do s1:=add(i[j], j=1..n); if s1 mod n = 0 then ans:=ans+1; fi; od: t0:=[op(t0), ans]; od:

MATHEMATICA

a[n_] := Sum[(-1)^(n+d)*EulerPhi[n/d]*Binomial[2d, d]/n, {d, Divisors[n]}]; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Oct 22 2012, after T. D. Noe's program in A145855 *)

PROG

(PARI) a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(n+d)*eulerphi(n/d)*binomial(2*d, d)/n)); \\ Altug Alkan, Aug 27 2018, after T. D. Noe at A145855

CROSSREFS

Cf. A061865, A119358, A145855, A318431, A318432, A318433.

Column k=2 of A304482.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jul 07 2010, based on a letter from Jean-Claude Babois.

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Aug 26 2018

STATUS

approved

A318432

Number of n-element subsets of [4n] whose elements sum to a multiple of n.

+10
4

1, 4, 12, 76, 460, 3104, 22404, 169152, 1315020, 10460416, 84764512, 697212652, 5805722692, 48847196896, 414623627136, 3546272614976, 30532934225100, 264420681260336, 2301782759539392, 20129523771781288, 176765807152990560, 1558058796052048968

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..21.

Marko Riedel et al., Number of n-element subsets divisible by n

FORMULA

a(n) = (-1)^n * (1/n) * Sum_{d|n} C(4d,d)*(-1)^d*phi(n/d) for n>0, a(0)=1.

MAPLE

with(numtheory); a := n -> `if`(n=0, 1, (-1)^n * 1/n * add(binomial(4*d, d)*(-1)^d*phi(n/d), d in divisors(n)));

PROG

(PARI) a(n) = if (n, (-1)^n * (1/n) * sumdiv(n, d, binomial(4*d, d)*(-1)^d*eulerphi(n/d)), 1); \\ Michel Marcus, Aug 27 2018

CROSSREFS

Cf. A169888, A318431, A318433.

Column k=4 of A304482.

KEYWORD

nonn

AUTHOR

Marko Riedel, Aug 26 2018

STATUS

approved

A318433

Number of n-element subsets of [5n] whose elements sum to a multiple of n.

+10
4

1, 5, 20, 155, 1220, 10630, 98900, 960650, 9613700, 98462675, 1027222520, 10877596900, 116613287300, 1263159501180, 13803839298920, 152000845788280, 1684888825463940, 18785707522181965, 210536007879090140, 2370423142929112065, 26799168520704093720

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..20.

Marko Riedel et al., Number of n-element subsets divisible by n

FORMULA

a(n) = (-1)^n * (1/n) * Sum_{d|n} C(5d,d)*(-1)^d*phi(n/d) for n>0, a(0)=1.

MAPLE

with(numtheory); a := n -> `if`(n=0, 1, (-1)^n * 1/n * add(binomial(5*d, d)*(-1)^d*phi(n/d), d in divisors(n)));

PROG

(PARI) a(n) = if (n, (-1)^n * (1/n) * sumdiv(n, d, binomial(5*d, d)*(-1)^d*eulerphi(n/d)), 1); \\ Michel Marcus, Aug 27 2018

CROSSREFS

Cf. A169888, A318431, A318432.

Column k=5 of A304482.

KEYWORD

nonn

AUTHOR

Marko Riedel, Aug 26 2018

STATUS

approved

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