A157674 -id:A157674 - cp's OEIS frontend

A137720 Expansion of sqrt(1-4x)/(1-3x).

1, 1, 1, -1, -13, -67, -285, -1119, -4215, -15505, -56239, -202309, -724499, -2589521, -9254363, -33111969, -118725597, -426892131, -1539965973, -5575175319, -20260052337, -73908397851, -270657727593, -994938310059

Offset: 0

Paul Barry, Feb 08 2008

Hankel transform is A120617. In general, sqrt(1-4*x)/(1-k*x) has Hankel transform with g.f. of (1-2*x)/(1+2*(k+2)*x+4*x^2).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A126966, A106191, A157674.

CoefficientList[Series[Sqrt[1-4*x]/(1-3*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 31 2014 *)
FullSimplify[Table[I*3^(-1/2+n) + 2^(1+2*n)*Gamma[1/2+n] * Hypergeometric2F1Regularized[1, 1/2+n, 2+n, 4/3]/(3*Sqrt[Pi]), {n, 0, 20}]] (* Vaclav Kotesovec, Jul 31 2014 *)

x='x+O('x^50); Vec(sqrt(1-4*x)/(1-3*x)) \\ G. C. Greubel, Mar 21 2017

a(n) = Sum_{k=0..n} 3^k*C(2*n-2*k,n-k)/(1-(2*n-2*k)).
D-finite with recurrence: n*a(n) + (6-7*n)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 16 2011
a(n) ~ -2^(2*n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 31 2014
a(n) = (-1)^n * A157674(2*n+1). - Vaclav Kotesovec, Jul 31 2014

A156909 G.f.: A(x) = 1 + xexp( Sum_{k>=1} [A(-(-1)^kx) - 1]^k/k ).

Original entry on oeis.org

1, 1, 1, 2, 3, 10, 18, 70, 135, 566, 1134, 4972, 10206, 46098, 96228, 443946, 938223, 4397730, 9382230, 44523232, 95698746, 458639492, 991787004, 4791683932, 10413763542, 50652087010, 110546105292, 540758574440, 1184422556700

Offset: 0

Paul D. Hanna, Mar 04 2009

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 10*x^5 + 18*x^6 + 70*x^7 + ...
...
A(x) = 1 + x*exp( [A(x)-1] + [A(-x)-1]^2/2 + [A(x)-1]^3/3 + [A(-x)-1]^4/4 + ...).

Cf. A156907, A156908.

Cf. A157674. - Paul D. Hanna, Mar 05 2009

{a(n)=local(A=1+x+x*O(x^n)); for(i=1,n,A=1+x*exp(-sum(k=1,n,(subst(A,x,(-1)^k*x+x*O(x^n))-1)^k/k))); polcoeff(A,n)}

{a(n)=local(B=(7-sqrt(1-12*x^2+x^2*O(x^n)))/6);polcoeff(B+sqrt(B^2-B),n)} \\ Paul D. Hanna, Mar 05 2009

From Paul D. Hanna, Mar 05 2009: (Start)
G.f.: A(x) = B(x) + sqrt(12*B(x) - 12 - 3*x^2)/3
where B(x) = (7-sqrt(1-12*x^2))/6 = A(x)*A(-x) = (A(x)+A(-x))/2 = 1 + x^2/(4-3*B(x)).
Lim_{n->infinity} a(2n)/a(2n-1) = 12^(1/3); lim_{n->infinity} a(2n+1)/a(2n) = 12^(2/3). (End)
D-finite with recurrence: 288*(n-6)*(n-5)*(n-4)*(n-3)*a(n-5) + 24*(n-4)*(n-3)*(52*n^2-378*n+761)*a(n-3) + 2*(n-1)*(181*n^3-271*n^2-950*n+1752)*a(n-1) - (n-1)*(n+1)*(87*n^2+38*n+48)*a(n+1) + 4*(n+1)*(n+2)*(n+3)*(n-1)*a(n+3) = 0. - Georg Fischer, Jul 15 2025

A229116 G.f.: A(x) = exp( Sum_{n>=1} A((-1)^nx)^n x^n/n ).

Original entry on oeis.org

1, 1, 0, 1, 0, 2, 2, 9, 8, 38, 28, 154, 126, 676, 602, 3129, 2816, 14718, 13384, 70334, 65204, 342108, 321788, 1686698, 1602214, 8402492, 8051652, 42239764, 40797750, 214045640, 208136494, 1092138905, 1068176200, 5606018286, 5511336912, 28929594902, 28571895096, 150000016044

Offset: 0

Paul D. Hanna, Sep 14 2013

nonn

Compare to a g.f. involving the Catalan function C(x) = 1 + x*C(x)^2 (A000108):

C(x) = exp( Sum_{n>=1} C(x)^n * x^n/n ).

			G.f.: A(x) = 1 + x + x^3 + 2*x^5 + 2*x^6 + 9*x^7 + 8*x^8 + 38*x^9 +...
where
log(A(x)) = A(-x)*x + A(x)^2*x^2/2 + A(-x)^3*x^3/3 + A(x)^4*x^4/4 + A(-x)^5*x^5/5 + A(x)^6*x^6/6 + A(-x)^7*x^7/7 +...
Also,
A(x)*(1 + x*A(x)) = 1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 6*x^6 + 18*x^7 + 30*x^8 + 76*x^9 + 124*x^10 + 308*x^11 + 514*x^12 +...
where 1/(A(x)*(1 + x*A(x))) = A(-x)*(1 - x*A(-x)).

Paul D. Hanna, Table of n, a(n) for n = 0..500
Vaclav Kotesovec, Recurrence (of order 14)

Cf. A157674.

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(k=1, n, subst(A, x, (-1)^k*x+x*O(x^n))^k*x^k/k))); polcoeff(A, n)}
for(n=0,50,print1(a(n),", "))

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sqrt( (1 + x*subst(A,x,-x))/((1 - x*subst(A,x,-x))*(1 - x^2*A^2)) +x*O(x^n))); polcoeff(A, n)}
for(n=0,50,print1(a(n),", "))

G.f. A(x) satisfies:
(1) A(x) = sqrt( (1 - x^2*A(-x)^2)/(1 - x^2*A(x)^2) ) / (1 - x*A(-x)).
(2) A(x) = 1/( (1 + x*A(x)) * A(-x) * (1 - x*A(-x)) ).
(3) 1 + x*A(x) = 2 / (1 + A(-x)^2*(1 - x^2*A(-x)^2)).
(4) A(x) = 1/(2*A(-x)*(1 - x*A(-x))) + A(-x)*(1 + x*A(-x))/2.
a(n) ~ c * d^n/(sqrt(Pi)*n^(3/2)), where d = sqrt((37 + (182701 - 19488*sqrt(87))^(1/3) + (182701 + 19488*sqrt(87))^(1/3))/21) = 2.37234975879070748... is the root of the equation -256 + 32*d^2 - 37*d^4 + 7*d^6 = 0. If n is even then c = sqrt((522 - 19*174^(2/3)/(92133 - 9877*sqrt(87))^(1/3) - (174*(92133 - 9877*sqrt(87)))^(1/3))/1479) = 0.3620905463490063953... is the root of the equation 182*c^2 - 522*c^4 + 493*c^6 = 16. If n is odd then c = sqrt(((58*(29 - 3*sqrt(87)))^(1/3) + (58*(29 + 3*sqrt(87)))^(1/3))/29) = 0.8049267655440167596... is the root of the equation 29*c^6 - 6*c^2 = 4. - Vaclav Kotesovec, Sep 15 2013

A277604 Array of coefficients T(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = sqrt(1 + 2xA(k,x) + (4k+1)x^2*(A(k,x))^2), k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 1, 7, 9, 13, 1, 1, 1, 9, 13, 37, 25, 1, 1, 1, 11, 17, 73, 81, 61, 1, 1, 1, 13, 21, 121, 169, 301, 125, 1, 1, 1, 15, 25, 181, 289, 841, 729, 295, 1, 1, 1, 17, 29, 253, 441, 1801, 2197, 2549, 625, 1, 1, 1, 19, 33, 337, 625, 3301, 4913, 10123, 6561, 1447, 1

Offset: 0

Werner Schulte, Oct 29 2016

For k = 0 see A000012, for k = 1 see A098615, and for k = 2 see A200376.
It will be interesting using the formulae for k < 0 (attention: signed terms!). Especially for k = -1 see A157674.
If G is the g.f. of central binomial coefficients (see A000984) and B(k,x) = G(k*x^2), then B(k,x) = A(k,x)/(1+x*A(k,x)) and A(k,x) = B(k,x) / (1-x*B(k,x)) for k >= 0. - Werner Schulte, Aug 07 2017

			The terms define the array T(k,n) for k >= 0 and n >= 0, i.e.,
k\n  0  1   2   3    4     5      6      7       8        9  . . .
0:   1  1   1   1    1     1      1      1       1        1  . . .
1:   1  1   3   5   13    25     61    125     295      625  . . .
2:   1  1   5   9   37    81    301    729    2549     6561  . . .
3:   1  1   7  13   73   169    841   2197   10123    28561  . . .
4:   1  1   9  17  121   289   1801   4913   28057    83521  . . .
5:   1  1  11  21  181   441   3301   9261   63071   194481  . . .
6:   1  1  13  25  253   625   5461  15625  123565   390625  . . .
7:   1  1  15  29  337   841   8401  24389  219619   707281  . . .
8:   1  1  17  33  433  1089  12241  35937  362993  1185921  . . .
9:   1  1  19  37  541  1369  17101  50653  567127  1874161  . . .
etc.

Cf. A000012, A000045, A000108, A000984, A015441, A046521, A098615, A111959, A200376.

A(k,x) = (x + sqrt(1 - 4*k*x^2))/(1 - (4*k+1)*x^2) for k >= 0.
T(k,0) = 1 and T(k,2*n+2) = (4*k+1)^(n+1)-2*(Sum_{i=0..n} A000108(i)*k^(i+1)* (4*k+1)^(n-i)), and T(k,2*n+1) = (4*k+1)^n for k >= 0 and n >= 0.
A(k,x) = 1/(1 - x - 2*k*x^2*C(k*x^2)), k >= 0, where C is the g.f. of A000108.
Conjecture: If B(k,n) satisfy B(k,0) = B(k,1) = 1 and B(k,n+2) = B(k,n+1) + k*B(k,n) for k >= 0 and n >= 0 (generalized Fibonacci numbers, see A015441) and G(k,x) = Sum_{n>=0} A000108(n)*B(k,n)*x^n for k >= 0, then you will have (1): A(k,x*G(k,x)) = G(k,x) and (2): G(k,x/A(k,x)) = A(k,x) for k >= 0. Especially for k = 1 see A098615 and for k = 2 see A200376.
Conjecture: T(k,2*n) = Sum_{i=0..n} A046521(n,i)*k^(n-i) for k, n >= 0. - Werner Schulte, Aug 02 2017
Recurrence: T(k,2*n+2) = (4*k+1)*T(k,2*n)-2*k^(n+1)*A000108(n) with initial value T(k,0) = 1 for k >= 0 and n >= 0. - Werner Schulte, Aug 09 2017
T(k,n) = Sum_{i=0..n} A111959(n,i)*k^((n-i)/2) for k >= 0 and n >= 0. - Werner Schulte, Aug 09 2017

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A137720 Expansion of sqrt(1-4*x)/(1-3*x).

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A156909 G.f.: A(x) = 1 + x*exp( Sum_{k>=1} [A(-(-1)^k*x) - 1]^k/k ).

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A229116 G.f.: A(x) = exp( Sum_{n>=1} A((-1)^n*x)^n * x^n/n ).

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A277604 Array of coefficients T(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = sqrt(1 + 2*x*A(k,x) + (4*k+1)*x^2*(A(k,x))^2), k >= 0.

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A137720 Expansion of sqrt(1-4x)/(1-3x).

A156909 G.f.: A(x) = 1 + xexp( Sum_{k>=1} [A(-(-1)^kx) - 1]^k/k ).

A229116 G.f.: A(x) = exp( Sum_{n>=1} A((-1)^nx)^n x^n/n ).

A277604 Array of coefficients T(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = sqrt(1 + 2xA(k,x) + (4k+1)x^2*(A(k,x))^2), k >= 0.