A229050 - OEIS

%I #15 Sep 25 2013 19:49:30

%S 1,2,9,1714,63079895,623361815288736,2670177752844538217570947,

%T 7363615666255986180456959666126927684,

%U 18165723931631174937747337664794705661513150850379149,53130688706387570972824498004857476332107293478561950967662962585645710

%N G.f.: Sum_{n>=0} (n^2)!/n!^n * x^n / (1-x)^(n^2+1).

%H Alois P. Heinz, <a href="/A229050/b229050.txt">Table of n, a(n) for n = 0..26</a>

%F a(n) = Sum_{k=0..n} Product_{j=0..k-1} binomial(n+j*k,k).

%F a(n) ~ exp(-1/12) * n^(n^2-n/2+1) / (2*Pi)^((n-1)/2). - _Vaclav Kotesovec_, Sep 23 2013

%e G.f.: A(x) = 1 + 2*x + 9*x^2 + 1714*x^3 + 63079895*x^4 + 623361815288736*x^5 +...

%e where

%e A(x) = 1/(1-x) + x/(1-x)^2 + (4!/2!^2)*x^2/(1-x)^5 + (9!/3!^3)*x^3/(1-x)^10 + (16!/4!^4)*x^4/(1-x)^17 + (25!/5!^5)*x^5/(1-x)^26 +...

%e Equivalently,

%e A(x) = 1/(1-x) + x/(1-x)^2 + 6*x^2/(1-x)^5 + 1680*x^3/(1-x)^10 + 63063000*x^4/(1-x)^17 + 623360743125120*x^5/(1-x)^26 +...+ A034841(n)*x^n/(1-x)^(n^2+1) +...

%e Illustrate formula a(n) = Sum_{k=0..n} Product_{j=0..k-1} C(n+j*k,k) for initial terms:

%e a(0) = 1;

%e a(1) = 1 + C(1,1);

%e a(2) = 1 + C(2,1) + C(2,2)*C(4,2);

%e a(3) = 1 + C(3,1) + C(3,2)*C(5,2) + C(3,3)*C(6,3)*C(9,3);

%e a(4) = 1 + C(4,1) + C(4,2)*C(6,2) + C(4,3)*C(7,3)*C(10,3) + C(4,4)*C(8,4)*C(12,4)*C(16,4);

%e a(5) = 1 + C(5,1) + C(5,2)*C(7,2) + C(5,3)*C(8,3)*C(11,3) + C(5,4)*C(9,4)*C(13,4)*C(17,4) + C(5,5)*C(10,5)*C(15,5)*C(20,5)*C(25,5); ...

%e which numerically equals:

%e a(0) = 1;

%e a(1) = 1 + 1 = 2;

%e a(2) = 1 + 2 + 1*6 = 9;

%e a(3) = 1 + 3 + 3*10 + 1*20*84 = 1714;

%e a(4) = 1 + 4 + 6*15 + 4*35*120 + 1*70*495*1820 = 63079895;

%e a(5) = 1 + 5 + 10*21 + 10*56*165 + 5*126*715*2380 + 1*252*3003*15504*53130 = 623361815288736; ...

%p with(combinat):

%p a:= n-> add(multinomial(n+(k-1)*k, n-k, k$k), k=0..n):

%p seq(a(n), n=0..15);  # _Alois P. Heinz_, Sep 23 2013

%t Table[Sum[Product[Binomial[n+j*k,k],{j,0,k-1}],{k,0,n}],{n,0,10}] (* _Vaclav Kotesovec_, Sep 23 2013 *)

%o (PARI) {a(n)=polcoeff(sum(m=0, n, (m^2)!/m!^m*x^m/(1-x+x*O(x^n))^(m^2+1)), n)}

%o for(n=0,15,print1(a(n),", "))

%o (PARI) {a(n)=sum(k=0,n,prod(j=0,k-1,binomial(n+j*k,k)))}

%o for(n=0,15,print1(a(n),", "))

%Y Cf. A229051, A034841; A001850, A081798, A082488, A082489, A229049.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 22 2013