A351856 Number of nonnegative integer solutions to 2*n = x_1 + x_2 + ... + x_n + 2*y_1 + 2*y_2 + ... + 2*y_n.

2, 14, 119, 1086, 10252, 98735, 963832, 9502014, 94386908, 943206264, 9471346755, 95491466655, 966026045376, 9800968460024, 99685873633744, 1016118049037630, 10377363759903252, 106161722891946356, 1087696666197827374, 11159365823946907336, 114631982782490824420

Offset

1,1

Comments

This is a companion sequence to A348410.

Suppose 2*n identical objects are distributed in 2*n labeled baskets, n colored white and n colored black. White baskets can contain any number of objects (or be empty), while black baskets must contain an even number of objects (or be empty). a(n) is the number of distinct possible distributions.

References

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Links

Table of n, a(n) for n=1..21.

Formula

a(n) = [x^(2*n)] ( 1/((1 - x)*(1 - x^2)) )^n.

a(n) = Sum_{k = 0..n} C(3*n-2*k-1,2*n-2*k)*C(n+k-1,k).

a(n) = Sum_{k = 0..2*n} (-1)^k*C(4*n-k-1,2*n-k)*C(n+k-1,k).

32*n*(n-1)*(2*n-1)*(2*n-3)*(41*n^2-126*n+93)*a(n) = 2*(n-1)*(2*n-3)*(16851*n^4-68637*n^3+93680*n^2-49024*n+7680)*a(n-1) - 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(41*n^2-44*n+8)*a(n-2) with a(1) = 2 and a(2) = 14.

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k.

Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and positive integers n and k.

The o.g.f. A(x) = 2*x + 14*x^2 + 119*x^3 + ... is the diagonal of the bivariate rational function x*t*(x - 1)*((x - 1)^2 - t)/((x - 1)^3 - t*(2*x + t - 2)) and hence is an algebraic function over Q(x) by Stanley 1999, Theorem 6.33, p. 197.

Let F(x) = (1/x)*Series_Reversion( x*sqrt((1-x)*(1-x^2)) ) and put G(x) = x*(d/dx)(log(F(x))). Then A(x^2) = (G(x) + G(-x))/2.

Example

n = 2: 14 distributions of 4 identical objects in 2 white and 2 black baskets
        White     Black
   1)  (0) (0)   [4] [0]
   2)  (0) (0)   [0] [4]
   3)  (0) (0)   [2] [2]
   4)  (2) (0)   [2] [0]
   5)  (0) (2)   [2] [0]
   6)  (1) (1)   [2] [0]
   7)  (2) (0)   [0] [2]
   8)  (0) (2)   [0] [2]
   9)  (1) (1)   [0] [2]
  10)  (4) (0)   [0] [0]
  11)  (0) (4)   [0] [0]
  12)  (3) (1)   [0] [0]
  13)  (1) (3)   [0] [0]
  14)  (2) (2)   [0] [0]

Maple

seq( add(binomial(3*n-2*k-1, 2*n-2*k)*binomial(n+k-1, k), k = 0..n), n = 1..20);

Crossrefs

Cf. A234839, A348410, A351857.

Sequence in context: A370671 A361962 A192457 * A155728 A267906 A199560

Adjacent sequences: A351853 A351854 A351855 * A351857 A351858 A351859

Keyword

nonn,easy

Author

Peter Bala, Feb 22 2022

Status

approved