A351857 Number of nonnegative integer solutions to n = x_1 + x_2 + ... + x_(2*n) + 2*y_1 + 2*y_2 + ... + 2*y_(2*n).

2, 14, 92, 654, 4752, 35204, 264112, 2000526, 15264866, 117161264, 903533380, 6995547780, 54343476072, 423360920528, 3306313730592, 25876855432846, 202909132368942, 1593755466338030, 12537009118650016, 98753463725849904, 778825917274945408, 6149069826564738780

Offset

1,1

Comments

This is a companion sequence to A348410.

Suppose n identical objects are distributed in 4*n labeled baskets, 2*n colored white and 2*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..22.

Formula

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

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

a(n) = Sum_{k = 0..n} (-1)^k*C(5*n-k-1,n-k)*C(2*n+k-1,k).

1024*n*(n-1)*(2*n-1)*(2*n-3)*(4*n-1)*(4*n-3)*P(n-1)*a(n) = 8*(n-1)*(2*n-3)*Q(n)*a(n-1) + 7*(7*n-8)*(7*n-9)*(7*n-10)*(7*n-11)*(7*n-12)*(7*n-13)*P(n)*a(n-2), with a(1) = 2, a(2) = 14, P(n) = 1744*n^4-3815*n^3+ 2920*n^2-912*n+96 and Q(n) = 46599680*n^8-381534880*n^7+1306363456*n^6- 2428492279*n^5+2661904813*n^4 -1747232452*n^3+664205312*n^2- 132046848*n+10321920.

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 + 92*x^3 + ... is the diagonal of the bivariate rational function x*t/(1 - t/((1 - x)*(1 - x^2))^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)^2*(1 - x^2)^2) ). Then A(x) = x*d/dx(Log(F(x)).

Example

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

Maple

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

Prog

(PARI) a(n) = sum(k = 0, n\2, binomial(3*n-2*k-1, n-2*k)*binomial(2*n+k-1, k)); \\ Michel Marcus, Feb 27 2022

Crossrefs

Cf. A234839, A348410, A351856.

Sequence in context: A020063 A323561 A288470 * A341395 A072148 A297474

Adjacent sequences: A351854 A351855 A351856 * A351858 A351859 A351860

Keyword

nonn,easy

Author

Peter Bala, Feb 22 2022

Status

approved