OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..26
FORMULA
a(n) = Sum_{k=0..n} 2^k * Product_{j=0..k-1} binomial(n+j*k,k).
a(n) ~ exp(-1/12) * n^(n^2-n/2+1) * 2^n / (2*Pi)^((n-1)/2). - Vaclav Kotesovec, Sep 23 2013
EXAMPLE
G.f.: A(x) = 1 + 3*x + 29*x^2 + 13567*x^3 + 1009142769*x^4 +...
where
A(x) = 1/(1-x) + (2*x)/(1-x)^2 + (4!/2!^2)*(2*x)^2/(1-x)^5 + (9!/3!^3)*(2*x)^3/(1-x)^10 + (16!/4!^4)*(2*x)^4/(1-x)^17 + (25!/5!^5)*(2*x)^5/(1-x)^26 +...
Equivalently,
A(x) = 1/(1-x) + (2*x)/(1-x)^2 + 6*(2*x)^2/(1-x)^5 + 1680*(2*x)^3/(1-x)^10 + 63063000*(2*x)^4/(1-x)^17 + 623360743125120*(2*x)^5/(1-x)^26 +...+ A034841(n)*(2*x)^n/(1-x)^(n^2+1) +...
Illustrate formula a(n) = Sum_{k=0..n} 2^k * Product_{j=0..k-1} C(n+j*k,k) for initial terms:
a(0) = 1;
a(1) = 1 + 2*C(1,1);
a(2) = 1 + 2*C(2,1) + 4*C(2,2)*C(4,2);
a(3) = 1 + 2*C(3,1) + 4*C(3,2)*C(5,2) + 8*C(3,3)*C(6,3)*C(9,3);
a(4) = 1 + 2*C(4,1) + 4*C(4,2)*C(6,2) + 8*C(4,3)*C(7,3)*C(10,3) + 16*C(4,4)*C(8,4)*C(12,4)*C(16,4);
a(5) = 1 + 2*C(5,1) + 4*C(5,2)*C(7,2) + 8*C(5,3)*C(8,3)*C(11,3) + 16*C(5,4)*C(9,4)*C(13,4)*C(17,4) + 32*C(5,5)*C(10,5)*C(15,5)*C(20,5)*C(25,5); ...
which numerically equals:
a(0) = 1;
a(1) = 1 + 2*1 = 3;
a(2) = 1 + 2*2 + 4*1*6 = 29;
a(3) = 1 + 2*3 + 4*3*10 + 8*1*20*84 = 13567;
a(4) = 1 + 2*4 + 4*6*15 + 8*4*35*120 + 16*1*70*495*1820 = 1009142769;
a(5) = 1 + 2*5 + 4*10*21 + 8*10*56*165 + 16*5*126*715*2380 + 32*1*252*3003*15504*53130 = 19947560933879891; ...
MAPLE
with(combinat):
a:= n-> add(2^k*multinomial(n+(k-1)*k, n-k, k$k), k=0..n):
seq(a(n), n=0..10); # Alois P. Heinz, Sep 23 2013
MATHEMATICA
Table[Sum[2^k*Product[Binomial[n+j*k, k], {j, 0, k-1}], {k, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, Sep 23 2013 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, (m^2)!/m!^m*(2*x)^m/(1-x+x*O(x^n))^(m^2+1)), n)}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, 2^k*prod(j=0, k-1, binomial(n+j*k, k)))}
for(n=0, 15, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 22 2013
STATUS
approved