A294640 G.f. A(x) = Sum_{n>=0} x^n/a(n) satisfies: A(x) = A(x^2) + Integral A(x^2) dx.

1, 1, 1, 3, 1, 5, 3, 21, 1, 9, 5, 55, 3, 39, 21, 315, 1, 17, 9, 171, 5, 105, 55, 1265, 3, 75, 39, 1053, 21, 609, 315, 9765, 1, 33, 17, 595, 9, 333, 171, 6669, 5, 205, 105, 4515, 55, 2475, 1265, 59455, 3, 147, 75, 3825, 39, 2067, 1053, 57915, 21, 1197, 609, 35931, 315, 19215, 9765, 615195, 1, 65, 33, 2211, 17, 1173, 595, 42245, 9, 657, 333, 24975, 171, 13167, 6669, 526851, 5, 405, 205, 17015, 105, 8925, 4515, 392805, 55, 4895, 2475, 225225, 1265, 117645, 59455, 5648225, 3, 291, 147, 14553, 75, 7575, 3825, 393975, 39, 4095, 2067, 221169, 1053, 114777, 57915, 6428565, 21, 2373, 1197, 137655, 609, 71253, 35931, 4275789, 315, 38115, 19215, 2363445, 9765, 1220625, 615195, 78129765, 1

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..4100

Formula

G.f. A(x) = Sum_{n>=0} x^n/a(n) satisfies:

(1) A'(x) = A(x^2) + 2*x*A'(x^2).

(2) A'(x) = A(x^2) + 2*x*A(x^4) + 4*x^3*A'(x^4).

(3) A'(x) = Sum_{n>=0} 2^n * x^(2^n-1) * A( x^(2^(n+1)) ).

(4) A(x) = 1 + Integral Sum_{n>=0} 2^n * x^(2^n-1) * A( x^(2^(n+1)) ) dx.

O.g.f. G(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) G(x) = G(x^2) + x * d/dx x*G(x^2).

(2) G(x) = (1+x)*G(x^2) + 2*x^3*G'(x^2).

a(2^n) = 1 for n>=0.

a(k*2^n) = a(k) for n>=0 and k>0.

a(2^n + 1) = 2^n + 1 for n>=1.

a(2^n - 1) = Product_{k=1..n} (2^k - 1) = A005329(n) for n>0.

a(3*2^n - 1) = Product_{k=1..n} (3*2^k - 1) for n>0.

a(m*2^n - 1) = Product_{k=1..n} (m*2^k - 1) for n>0 and positive odd m.

Limit_{n->oo} Sum_{k=0..2^n} 1/(a(k) * a(2^n-k)) = 3.9409369799444642172...

Example

G.f. A(x) = Sum_{n>=0} x^n/a(n) begins:
A(x) = 1/1 + x/1 + x^2/1 + x^3/3 + x^4/1 + x^5/5 + x^6/3 + x^7/21 + x^8/1 + x^9/9 + x^10/5 + x^11/55 + x^12/3 + x^13/39 + x^14/21 + x^15/315 + x^16/1 + x^17/17 + x^18/9 + x^19/171 + x^20/5 + x^21/105 + x^22/55 + x^23/1265 + x^24/3 + x^25/75 + x^26/39 + x^27/1053 + x^28/21 + x^29/609 + x^30/315 + x^31/9765 + x^32/1 + x^33/33 + x^34/17 + x^35/595 + x^36/9 + x^37/333 + x^38/171 + x^39/6669 + x^40/5 + x^41/205 + x^42/105 + x^43/4515 + x^44/55 + x^45/2475 + x^46/1265 + x^47/59455 + x^48/3 + x^49/147 + x^50/75 + x^51/3825 + x^52/39 + x^53/2067 + x^54/1053 + x^55/57915 + x^56/21 + x^57/1197 + x^58/609 + x^59/35931 + x^60/315 + x^61/19215 + x^62/9765 + x^63/615195 + x^64/1 +...+ x^n/a(n) +...
such that A(x) = A(x^2) + Integral A(x^2) dx.
Further,
A'(x) = A(x^2) + 2*x*A(x^4) + 4*x^3*A(x^8) + 8*x^7*A(x^16) + 16*x^15*A(x^32) + 32*x^31*A(x^64) +...+ 2^n * x^(2^n-1) * A( x^(2^(n+1)) ) +...
where A'(x) = A(x^2) + 2*x*A'(x^2).
RELATED SERIES.
A'(x) = 1/1 + 2*x/1 + x^2/1 + 4*x^3/1 + x^4/1 + 2*x^5/1 + x^6/3 + 8*x^7/1 + x^8/1 + 2*x^9/1 + x^10/5 + 4*x^11/1 + x^12/3 + 2*x^13/3 + x^14/21 + 16*x^15/1 + x^16/1 + 2*x^17/1 + x^18/9 + 4*x^19 + x^20/5 + 2*x^21/5 + x^22/55 + 8*x^23/1 + x^24/3 + 2*x^25/3 + x^26/39 + 4*x^27/3 + x^28/21 + 2*x^29/21 + x^30/315 + 32*x^31/1 + x^32/1 +...
Integral A(x^2) dx = x/1 + x^3/3 + x^5/5 + x^7/21 + x^9/9 + x^11/55 + x^13/39 + x^15/315 + x^17/17 + x^19/171 + x^21/105 + x^23/1265 + x^25/75 + x^27/1053 + x^29/609 + x^31/9765 + x^33/33 + x^35/595 + x^37/333 + x^39/6669 + x^41/205 + x^43/4515 + x^45/2475 + x^47/59455 + x^49/147 + x^51/3825 + x^53/2067 + x^55/57915 + x^57/1197 + x^59/35931 + x^61/19215 + x^63/615195 + x^65/65 +...
Also, we may write the g.f. as the series
A(x) = 1 + x + 2*x^2/2! + 2*x^3/3! + 24*x^4/4! + 24*x^5/5! + 240*x^6/6! + 240*x^7/7! + 40320*x^8/8! + 40320*x^9/9! + 725760*x^10/10! + 725760*x^11/11! + 159667200*x^12/12! + 159667200*x^13/13! + 4151347200*x^14/14! + 4151347200*x^15/15! + 20922789888000*x^16/16! + 20922789888000*x^17/17! + 711374856192000*x^18/18! + 711374856192000*x^19/19! + 486580401635328000*x^20/20! + 486580401635328000*x^21/21! + 20436376868683776000*x^22/22! + 20436376868683776000*x^23/23! +...+ n!/a(n) * x^n/n! +...
The terms at positions 2^n - 1 begin:
[1, 1, 3, 21, 315, 9765, 615195, 78129765, 19923090075, ..., A005329(n), ...].
The terms at positions 3*2^n - 1 begin:
[1, 5, 55, 1265, 59455, 5648225, 1078810975, 413184603425, 316912590826975, ...].

Prog

(PARI) {a(n) = my(A=1); for(i=1, #binary(n+1), A = subst(A, x, x^2) + intformal( subst(A, x, x^2) +x*O(x^n)) ); 1/polcoeff(A, n)}
for(n=0, 128, print1(a(n), ", "))

Crossrefs

Cf. A005329.

Sequence in context: A289094 A171382 A002323 * A365671 A200920 A290534

Adjacent sequences: A294637 A294638 A294639 * A294641 A294642 A294643

Keyword

nonn,look

Author

Paul D. Hanna, Nov 05 2017

Status

approved