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A245310 G.f. A(x) satisfies: [x^(n+1)] A(x)^n = n*([x^(n-1)] A(x)^n) for n>=1. 3
1, 1, 1, 1, 2, 4, 12, 34, 120, 412, 1608, 6244, 26288, 111448, 499256, 2265288, 10701896, 51339768, 254175048, 1278947304, 6604214760, 34662182904, 186002333640, 1014140252376, 5638617162312, 31837193871480, 182962292354376, 1067120997002680, 6325487157903240, 38030207563538680 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The g.f. G(x) of the closely related sequence A245311 satisfies:

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

(2) G(x)^2 = Dx(G(x)) / (1 + x^2*Dx^2(G(x))), where Dx(F(x)) = d/dx x*F(x).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..500

FORMULA

G.f. satisfies:

(1) A(x) = 1 + x + x^2 + x^3*A'(x)/(A(x) - x*A'(x)).

(2) A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A245311.

(3) A(x) = x / Series_Reversion(x*G(x)) where G(x) is the g.f. of A245311.

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 12*x^6 + 34*x^7 +...

The table of coefficients of x^k in A(x)^n begin:

n=1: [1, 1,  1,   1,   2,    4,   12,   34,   120,   412,   1608, ...];

n=2: [1, 2,  3,   4,   7,   14,   37,  104,   344,  1172,   4412, ...];

n=3: [1, 3,  6,  10,  18,   36,   88,  240,   753,  2515,   9174, ...];

n=4: [1, 4, 10,  20,  39,   80,  188,  496,  1487,  4836,  17122, ...];

n=5: [1, 5, 15,  35,  75,  161,  375,  965,  2785,  8795,  30241, ...];

n=6: [1, 6, 21,  56, 132,  300,  708, 1800,  5046, 15484,  51738, ...];

n=7: [1, 7, 28,  84, 217,  525, 1274, 3242,  8918, 26684,  86772, ...];

n=8: [1, 8, 36, 120, 338,  872, 2196, 5656, 15423, 45248, 143576, ...];

n=9: [1, 9, 45, 165, 504, 1386, 3642, 9576, 26127, 75655, 235143, ...]; ...

from which we can illustrate [x^(n+1)] A(x)^n = n*([x^(n-1)] A(x)^n):

n=1: [x^2] A(x) = 1 = 1*([x^0] A(x)) = 1*1 ;

n=2: [x^3] A(x)^2 = 4 = 2*([x^1] A(x)^2) = 2*2 ;

n=3: [x^4] A(x)^3 = 18 = 3*([x^2] A(x)^3) = 3*6 ;

n=4: [x^5] A(x)^4 = 80 = 4*([x^3] A(x)^4) = 4*20 ;

n=5: [x^6] A(x)^5 = 375 = 5*([x^4] A(x)^5) = 5*75 ;

n=6: [x^7] A(x)^6 = 1800 = 6*([x^5] A(x)^6) = 6*300 ;

n=7: [x^8] A(x)^7 = 8918 = 7*([x^6] A(x)^7) = 7*1274 ;

n=8: [x^9] A(x)^8 = 45248 = 8*([x^7] A(x)^8) = 8*5656 ;

n=9: [x^10] A(x)^9 = 235143 = 9*([x^8] A(x)^9) = 9*26127 ; ...

describing terms that lie along diagonals in the above table.

From the main diagonal in the above table, we may derive A245311:

[1/1, 2/2, 6/3, 20/4, 75/5, 300/6, 1274/7, 5656/8, 26127/9, ...]

= [1, 1, 2, 5, 15, 50, 182, 707, 2903, 12479, ...].

PROG

(PARI) /* From A(x) = 1+x+x^2 + x^3*A'(x)/(A(x) - x*A'(x)): */

{a(n)=local(A=1+x); for(i=1, n, A = 1+x+x^2 + x^3*A'/(A-x*A' +x*O(x^n))); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

(PARI) /* From [x^(n-1)] A(x)^n = n*([x^(n+1)] A(x)^n): */

{a(n)=local(A=1+x+x^2); for(m=2, n, A=A-x^(m+1)*(polcoeff(A^m+O(x^(m+2)), m+1)/m - polcoeff(A^m+O(x^m), m-1))+O(x^(n+2))); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

(PARI) /* From the g.f. of A245311: */

{a(n)=local(G=1+x); for(i=1, n, G = (1 + x^2*G^2 + x^3*G*G')/(1-x +x*O(x^n)));

polcoeff(x/serreverse(x*G +x^2*O(x^n)), n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A245311, A245312.

Sequence in context: A148202 A148203 A240904 * A215953 A209027 A069727

Adjacent sequences:  A245307 A245308 A245309 * A245311 A245312 A245313

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 17 2014

STATUS

approved

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Last modified May 24 02:39 EDT 2022. Contains 354000 sequences. (Running on oeis4.)