|
Search: id:A155200
|
|
|
| A155200 |
|
G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2) * x^n/n ), a power series in x with integer coefficients. |
|
+0
31
|
|
| 1, 2, 10, 188, 16774, 6745436, 11466849412, 80444398636280, 2306003967992402758, 268654794629082985019564, 126765597346260977505891041836, 241678070948246232010898235031930952
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
More generally, for m integer, exp( Sum_{n>=1} m^(n^2) * x^n/n ) is a power series in x with integer coefficients.
CONJECTURE: highest exponent of 2 dividing a(n) = A000120(n) = number of 1's in binary expansion of n, so that a(n)/2^A000120(n) is odd for n>=0. [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 01 2009]
|
|
FORMULA
|
Equals column 0 of triangle A155810.
G.f. satisfies: 2*A(x)*A(4x) + 8*x*A(x)*A'(4x) - A'(x)*A(4x) = 0. [From Paul D. Hanna (pauldhanna(AT)juno.com), Feb 24 2009]
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Mar 11 2009: (Start)
The differential equation implies recurrence:
n*a(n) = 2*a(n-1) + sum(k=1,n-1,4^k*a(k)*[2*(k+1)*a(n-1-k) - (n-k)*a(n-k)] for n>0, with a(0)=1.
G.f. A(x) generates A156631:
A156631(n) = [x^n] A(x)^(2^n) for n>=0, where the g.f. of A156631 = Sum_{n>=0} [Sum_{k>=1} (2^n*2^k*x)^k/k]^n/n!. (End)
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Sep 01 2009: (Start)
a(n) = (1/n)*Sum_{k=1..n} 2^(k^2)*a(n-k), a(0) = 1. [from Vladeta Jovovic, Feb 4 2009].
Euler transform of A159034. [from Vladeta Jovovic, Apr 2 2009].
(End)
|
|
EXAMPLE
|
G.f.: A(x) = 1 + 2*x + 10*x^2 + 188*x^3 + 16774*x^4 + 6745436*x^5 +...
log(A(x)) = 2*x + 2^4*x^2/2 + 2^9*x^3/3 + 2^16*x^4/4 + 2^25*x^5/5 +...
|
|
PROGRAM
|
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, 2^(m^2)*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=if(n==0, 1, (1/n)*(2*a(n-1) + sum(k=1, n-1, 4^k*a(k)*(2*(k+1)*a(n-1-k) - (n-k)*a(n-k)))))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Mar 11 2009]
(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, 2^(k^2)*a(n-k)))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 01 2009]
|
|
CROSSREFS
|
Cf. A155201, A155202, A155810 (triangle), variants: A155203, A155207.
Cf. A156631. [From Paul D. Hanna (pauldhanna(AT)juno.com), Mar 11 2009]
Cf. A000120; A159034. [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 01 2009]
Sequence in context: A086675 A057119 A037267 this_sequence A156510 A159558 A001528
Adjacent sequences: A155197 A155198 A155199 this_sequence A155201 A155202 A155203
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2009
|
|
|
Search completed in 0.002 seconds
|
