A318557 -id:A318557 - cp's OEIS frontend

A119358 Number of n-element subsets of [2n] having an even sum.

1, 1, 2, 10, 38, 126, 452, 1716, 6470, 24310, 92252, 352716, 1352540, 5200300, 20056584, 77558760, 300546630, 1166803110, 4537543340, 17672631900, 68923356788, 269128937220, 1052049129144, 4116715363800, 16123803193628, 63205303218876, 247959261273752

Offset: 0

Paul Barry, May 16 2006

Old name was: Central coefficients of number triangle A119326.

			a(3) = 10: {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {2,4,6}, {3,4,5}, {3,5,6}. - _Alois P. Heinz_, Feb 04 2017

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A071688, A119326, A119359, A169888, A282011.

Column k=2 of A318557.

a:= proc(n) option remember; `if`(n<3, 1+n*(n-1)/2,
     ((4*n-10)*(5*n^2-10*n+4)*(a(n-1)+4*(n-2)*a(n-3)
      /(n-1))/(5*n^2-20*n+19)-4*(n-1)*a(n-2))/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 26 2018

Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1/2, 1/2, 1}, 1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 04 2016 *)

G.f.: (1/sqrt(1-4x)+1/sqrt(1+4x^2))/2.
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)^2.
a(n) = C(2n,n)/2+sin(Pi*(n+1)/2)*C(n,n/2)/2.
a(n) = A119326(2n,n).
a(n) = A071688(n) + A119359(n) for n>=1.
D-finite with recurrence n*(n-1)*(10*n-29)*a(n) +2*(n-1)*(5*n^2-74*n+164)*a(n-1) +4*(-40*n^3+310*n^2 -744*n+559)*a(n-2) +8*(n-2)*(5*n^2-74*n+164)*a(n-3) -16*(25*n-42)*(n-3)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 05 2012
a(n) = hypergeom([(1-n)/2, (1-n)/2, -n/2, -n/2], [1/2, 1/2, 1], 1). - Vladimir Reshetnikov, Oct 04 2016
a(n) = A282011(2n,n). - Alois P. Heinz, Feb 04 2017

New name from Alois P. Heinz, Feb 04 2017

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.

Original entry on oeis.org

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

Offset: 0

Marko Riedel, Aug 28 2018

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.

			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, ...

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

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

Main diagonal gives A318477.

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);

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 *)

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

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.

A318477 Number of n-member subsets of [n^2] whose elements sum to a multiple of n.

Original entry on oeis.org

1, 1, 2, 30, 460, 10630, 324516, 12271518, 553275192, 28987537806, 1731030733840, 116068178638786, 8634941165110140, 705873715441872276, 62895036883536770108, 6067037854078500844740, 629921975126483973659888, 70043473196734767582082246

Offset: 0

Alois P. Heinz, Aug 26 2018

nonn

			a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 2: {1,3}, {2,4}.
a(3) = 30: {1,2,3}, {1,2,6}, {1,2,9}, {1,3,5}, {1,3,8}, {1,4,7}, {1,5,6}, {1,5,9}, {1,6,8}, {1,8,9}, {2,3,4}, {2,3,7}, {2,4,6}, {2,4,9}, {2,5,8}, {2,6,7}, {2,7,9}, {3,4,5}, {3,4,8}, {3,5,7}, {3,6,9}, {3,7,8}, {4,5,6}, {4,5,9}, {4,6,8}, {4,8,9}, {5,6,7}, {5,7,9}, {6,7,8}, {7,8,9}.

Alois P. Heinz, Table of n, a(n) for n = 0..338

Main diagonal of A304482 and of A318557.

Cf. A119358, A169888, A308667.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(phi(n/d)*
      (-1)^(n+d)*binomial(n*d, d), d=divisors(n))/n)
    end:
seq(a(n), n=0..20);

a[n_] := (-1)^n Sum[(-1)^d Binomial[d n, d] EulerPhi[n/d], {d, Divisors[n]} ]/n; a[0] = 1;
a /@ Range[0, 20] (* Jean-François Alcover, Sep 23 2019 *)

a(n) = n * A308667(n) for n >= 1.

a(n) ~ exp(n - 1/2) * n^(n - 3/2) / sqrt(2*Pi). - Vaclav Kotesovec, Mar 28 2023

A308667 (1/n) times the number of n-member subsets of [n^2] whose elements sum to a multiple of n.

Original entry on oeis.org

1, 1, 10, 115, 2126, 54086, 1753074, 69159399, 3220837534, 173103073384, 10551652603526, 719578430425845, 54297978110913252, 4492502634538340722, 404469190271900056316, 39370123445405248353743, 4120204305690280446004838, 461365717080848755611811094

Offset: 1

Alois P. Heinz, Jul 14 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..339
Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Main diagonal of A309148.

Cf. A304482, A318557, A318477.

with(numtheory):
a:= proc(n) option remember; add(phi(n/d)*
      (-1)^(n+d)*binomial(n*d, d), d=divisors(n))/n^2
    end:
seq(a(n), n=1..20);

a[n_] := a[n] = Sum[EulerPhi[n/d]*
     (-1)^(n + d)*Binomial[n*d, d], {d, Divisors[n]}]/n^2;
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 24 2022, after Alois P. Heinz *)

a(n) = A309148(n,n).
a(n) = (1/n) * A318477(n).
a(p) == 1 (mod p^3) for all primes p >= 5 (apply Meštrović, Remark 17, p. 12). - Peter Bala, Mar 28 2023
a(n) ~ exp(n - 1/2) * n^(n - 5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Mar 28 2023

A318591 Number of n-member subsets of [3*n] whose elements sum to a multiple of three.

Original entry on oeis.org

1, 1, 5, 30, 165, 1001, 6198, 38760, 245157, 1562331, 10015005, 64512240, 417226230, 2707475148, 17620076360, 114955810530, 751616304549, 4923689695575, 32308782871911, 212327989773900, 1397281501935165, 9206478467531865, 60727722660586800, 400978991944396320

Offset: 0

Alois P. Heinz, Aug 29 2018

nonn

			a(2) = 5: {1,2}, {1,5}, {2,4}, {3,6}, {4,5}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=3 of A318557.

A318592 Number of n-member subsets of [4*n] whose elements sum to a multiple of four.

Original entry on oeis.org

1, 1, 6, 55, 460, 3876, 33594, 296010, 2630044, 23535820, 211911256, 1917334783, 17417167156, 158753389900, 1451182694940, 13298522298180, 122131736900380, 1123787895356412, 10358022417860040, 95615237915961100, 883829035764950960, 8179808679272664720

Offset: 0

Alois P. Heinz, Aug 29 2018

nonn

			a(2) = 6: {1,3}, {1,7}, {2,6}, {3,5}, {4,8}, {5,7}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=4 of A318557.

A318593 Number of n-member subsets of [5*n] whose elements sum to a multiple of five.

Original entry on oeis.org

1, 1, 9, 91, 969, 10630, 118755, 1344904, 15380937, 177232627, 2054455670, 23930713170, 279871768995, 3284214703056, 38650751381832, 456002537343580, 5391644226101705, 63871405575418665, 757929628541719755, 9007607943130625829, 107196674080761940470

Offset: 0

Alois P. Heinz, Aug 29 2018

nonn

			a(2) = 9: {1,4}, {1,9}, {2,3}, {2,8}, {3,7}, {4,6}, {5,10}, {6,9}, {7,8}.

Alois P. Heinz, Table of n, a(n) for n = 0..922

Column k=5 of A318557.

Cf. A163455, A163456.

b:= proc(n, s, m, t) option remember; `if`(n=0, `if`(s=0 and t=0, 1, 0),
      b(n-1, s, m, t)+`if`(t=0, 0, b(n-1, irem(s+n, m), m, t-1)))
    end:
a:= n-> b(5*n, 0, 5, n):
seq(a(n), n=0..27);

a(n) = floor(A163456(n)) + [n mod 5 = 0]*A163455(n/5), with A163456(n) = binomial(5*n,n)/5 and A163455(n) = binomial(5*n-1,n) where [] is an Iverson bracket. - Georg Fischer, Mar 23 2019

A318594 Number of n-member subsets of [6*n] whose elements sum to a multiple of six.

Original entry on oeis.org

1, 1, 10, 138, 1782, 23751, 324516, 4496388, 62893270, 886322982, 12565647510, 179013799328, 2560547711772, 36749279048405, 528955735258632, 7632612327457638, 110375347460981718, 1599199100396803290, 23209746715943492412, 337362380452228509840

Offset: 0

Alois P. Heinz, Aug 29 2018

nonn

			a(2) = 10: {1,5}, {1,11}, {2,4}, {2,10}, {3,9}, {4,8}, {5,7}, {6,12}, {7,11}, {8,10}.

Alois P. Heinz, Table of n, a(n) for n = 0..853

Column k=6 of A318557.

A318595 Number of n-member subsets of [7*n] whose elements sum to a multiple of seven.

Original entry on oeis.org

1, 1, 13, 190, 2925, 46376, 749398, 12271518, 202927725, 3381098545, 56672074888, 954526728530, 16141841823510, 273897571557780, 4660989191504198, 79515468511191440, 1359472297966470765, 23287391568864135063, 399584727647019644125, 6866816177157320098000

Offset: 0

Alois P. Heinz, Aug 29 2018

nonn

			a(2) = 13: {1,6}, {1,13}, {2,5}, {2,12}, {3,4}, {3,11}, {4,10}, {5,9}, {6,8}, {7,14}, {8,13}, {9,12}, {10,11}.

Alois P. Heinz, Table of n, a(n) for n = 0..804

Column k=7 of A318557.

A318596 Number of n-member subsets of [8*n] whose elements sum to a multiple of eight.

Original entry on oeis.org

1, 1, 14, 253, 4508, 82251, 1533686, 28989675, 553275192, 10639125640, 205811431514, 4000751045226, 78083583349868, 1529143625215220, 30032524962905850, 591315394579074378, 11667877650711051896, 230672804560960311000, 4568131162533134953328

Offset: 0

Alois P. Heinz, Aug 29 2018

nonn

			a(2) = 14: {1,7}, {1,15}, {2,6}, {2,14}, {3,5}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}, {8,16}, {9,15}, {10,14}, {11,13}.

Alois P. Heinz, Table of n, a(n) for n = 0..765

Column k=8 of A318557.

cp's OEIS Frontend

A119358 Number of n-element subsets of [2n] having an even sum.

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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.

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A318477 Number of n-member subsets of [n^2] whose elements sum to a multiple of n.

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A308667 (1/n) times the number of n-member subsets of [n^2] whose elements sum to a multiple of n.

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A318591 Number of n-member subsets of [3*n] whose elements sum to a multiple of three.

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A318592 Number of n-member subsets of [4*n] whose elements sum to a multiple of four.

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A318593 Number of n-member subsets of [5*n] whose elements sum to a multiple of five.

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A318594 Number of n-member subsets of [6*n] whose elements sum to a multiple of six.

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A318595 Number of n-member subsets of [7*n] whose elements sum to a multiple of seven.