A317354 G.f. satisfies: A(x) = 1 + x*A(x) * ( d/dx x*A(x)^4 ).

1, 1, 9, 143, 3077, 81186, 2500555, 87388600, 3402833181, 145793616851, 6809846205490, 344295007127226, 18732966469820743, 1091592281265899216, 67839235267146659208, 4479984759698760684608, 313349016085369145384269, 23145049153829729002326633, 1800552901345736894764054891, 147168026141415485071183102785

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..362

Formula

O.g.f. A(x) satisfies:

(1) A(x) = 1 + x*A(x) * ( d/dx x*A(x)^4 ).

(2) [x^n] exp( n * x*A(x)^4 ) / A(x) = 0 for n>0.

(3.a) [x^n] exp(-n * x*A(x)^4) * (2 - 1/A(x)) = 0 for n >= 1.

(3.b) [x^n] exp(-n^2 * x*A(x)^4) * (n + 1 - n/A(x)) = 0 for n >= 1.

(3.c) [x^n] exp(-n^(p+1) * x*A(x)^4) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0.

a(n) ~ c * 4^n * n! * n^(5/4), where c = 0.1196141591155152938065262... - Vaclav Kotesovec, Aug 09 2018

Example

O.g.f.: A(x) = 1 + x + 9*x^2 + 143*x^3 + 3077*x^4 + 81186*x^5 + 2500555*x^6 + 87388600*x^7 + 3402833181*x^8 + ...
where A(x) = 1 + x*A(x)^5 + 4*x^2*A(x)^4*A'(x).
RELATED TABLE.
The table of coefficients of x^k/k! in exp( n*x*A(x)^4 ) / A(x) begins:
n=1: [1, 0, -9, -554, -52995, -7440324, -1434977405, ...]
n=2: [1, 1, 0, -304, -38032, -5920008, -1205232128, ...];
n=3: [1, 2, 11, 0, -20475, -4189122, -949414689, ...];
n=4: [1, 3, 24, 364, 0, -2224296, -665082368, ...];
n=5: [1, 4, 39, 794, 23741, 0, -349575245, ...];
n=6: [1, 5, 56, 1296, 51120, 2511576, 0, ...];
n=7: [1, 6, 75, 1876, 82533, 5340642, 386787007, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + x*A*(x*A^4)' +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A-1; A[#A] = -Vec( exp(-m * x*Ser(A)^4 ) * (2 - 1/Ser(A)))[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A-1; A[#A] = -Vec( exp(-m^2 * x*Ser(A)^4 ) * (m+1 - m/Ser(A)))[#A]/m ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A088716, A317352, A317353.

Sequence in context: A187402 A363478 A303147 * A050787 A017198 A281789

Adjacent sequences: A317351 A317352 A317353 * A317355 A317356 A317357

Keyword

nonn

Author

Paul D. Hanna, Jul 26 2018

Status

approved