This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A352373 #20 Dec 26 2024 04:21:46
%S A352373 2,12,74,484,3252,22260,154352,1080612,7621526,54071512,385454940,
%T A352373 2758690636,19810063392,142662737376,1029931873824,7451492628260,
%U A352373 54013574117106,392188079586468,2851934621212598,20766924805302984,151403389181347160,1105047483656041080
%N A352373 a(n) = [x^n] ( 1/((1 - x)^2*(1 - x^2)) )^n for n >= 1.
%C A352373 Suppose n identical objects are distributed in 3*n labeled baskets, 2*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.
%C A352373 Number of nonnegative integer solutions to n = x_1 + x_2 + ... + x_(2*n) + 2*y_1 + 2*y_2 + ... + 2*y_n.
%C A352373 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.
%C A352373 Calculation suggests that, in fact, stronger congruences may hold.
%C A352373 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.
%C A352373 More generally, let r and s be integers and define a sequence (a(r,s;n))n>=1 by a(r,s;n) = [x^n] ( (1 + x)^r * (1 - x)^s )^n.
%C A352373 Conjecture: for each r and s the above supercongruences hold for the sequence (a(r,s;n))n>=1.
%C A352373 The present sequence is the case r = -1 and s = -3. Other cases include A000984 (r = 2, s = 0), A001700 with offset 1 (r = 0, s = -1), A002003 (r = 1, s = -1), A091527 (r = 3, s = -1), A119259 (r = 2, s = -1), A156894 (r = 1, s = -2), A165817 (r = 0, s = -2), A234839 (r = 1, s = 2), A348410 (r = -1, s = -2) and A351857 (r = -2, s = -4).
%D A352373 R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
%H A352373 Paolo Xausa, <a href="/A352373/b352373.txt">Table of n, a(n) for n = 1..1000</a>
%F A352373 a(n) = Sum_{k = 0..floor(n/2)} binomial(3*n-2*k-1,n-2*k)*binomial(n+k-1,k).
%F A352373 a(n) = Sum_{k = 0..n} (-1)^k*binomial(4*n-k-1,n-k)*binomial(n+k-1,k).
%F A352373 a(n) = binomial(4*n-1,n)*hypergeom([n, -n], [1-4*n], -1).
%F A352373 48*n*(n-1)*(3*n-1)*(3*n-2)*(93*n^3-434*n^2+668*n-339)*a(n) = 12*(n-1)*(21762*n^6-134199*n^5+323805*n^4-386685*n^3+237728*n^2-70336*n+7680)*a(n-1) + 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(93*n^3-155*n^2+79*n-12)*a(n-2) with a(1) = 2 and a(2) = 12.
%F A352373 The o.g.f. A(x) = 2*x + 12*x^2 + 74*x^3 + ... is the diagonal of the bivariate rational function x*t/(1 - t/((1 - x)^2*(1 - x^2))) and hence is an algebraic function over Q(x) by Stanley 1999, Theorem 6.33, p. 197.
%F A352373 A(x) = x*d/dx(log(F(x))), where F(x) = (1/x)*Series_Reversion( x*(1 - x)^2*(1 - x^2) ).
%F A352373 a(n) ~ sqrt(4 + sqrt(6)) * (13/4 + 31*sqrt(6)/18)^n / (2*sqrt(5*Pi*n)). - _Vaclav Kotesovec_, Mar 15 2022
%e A352373 n = 2: 12 distributions of 2 identical objects in 4 white and 2 black baskets
%e A352373 White Black
%e A352373 1) (0) (0) (0) (0) [2] [0]
%e A352373 2) (0) (0) (0) (0) [0] [2]
%e A352373 3) (2) (0) (0) (0) [0] [0]
%e A352373 4) (0) (2) (0) (0) [0] [0]
%e A352373 5) (0) (0) (2) (0) [0] [0]
%e A352373 6) (0) (0) (0) (2) [0] [0]
%e A352373 7) (1) (1) (0) (0) [0] [0]
%e A352373 8) (1) (0) (1) (0) [0] [0]
%e A352373 9) (1) (0) (0) (1) [0] [0]
%e A352373 10) (0) (1) (1) (0) [0] [0]
%e A352373 11) (0) (1) (0) (1) [0] [0]
%e A352373 12) (0) (0) (1) (1) [0] [0]
%e A352373 Examples of supercongruences:
%e A352373 a(7) - a(1) = 154352 - 2 = 2*(3^2)*(5^2)*(7^3) == 0 (mod 7^3);
%e A352373 a(2*11) - a(2) = 1105047483656041080 - 12 = (2^2)*3*(11^3)*13*101*103*2441* 209581 == 0 (mod 11^3).
%p A352373 seq(add( binomial(3*n-2*k-1,n-2*k)*binomial(n+k-1,k), k = 0..floor(n/2)), n = 1..25);
%t A352373 nterms=25;Table[Sum[Binomial[3n-2k-1,n-2k]Binomial[n+k-1,k],{k,0,Floor[n/2]}],{n,nterms}] (* _Paolo Xausa_, Apr 10 2022 *)
%Y A352373 Cf. A000984, A001448, A001700, A002003, A091527, A119259, A156894, A165817, A211419, A211421, A234839, A262733, A276098, A348410, A351856, A351857.
%K A352373 nonn,easy
%O A352373 1,1
%A A352373 _Peter Bala_, Mar 14 2022