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A232695 E.g.f. A(x) satisfies: A'(x) = A(x*A'(x)^5) with A(0)=1. 8
1, 1, 1, 11, 266, 10326, 562926, 40058076, 3554828286, 381374161166, 48366170807276, 7128626213386476, 1204840675597360776, 230986547885416953936, 49777541426984300127816, 11964954349177005321013976, 3186498480002528225295506276, 934756070179948684556233837476 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

CONJECTURES.

a(n) == 1 (mod 5) for n>=0.

a(n) == 0 (mod 2) for n>=4.

a(n) == 0 (mod 2^2) for n>=10.

a(n) == 0 (mod 2^3) for n>=18.

a(n) == 0 (mod 2^k) for n>=(8*k-6) for k>1.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..100

FORMULA

E.g.f. satisfies: A(x) = A'(x/A(x)^5).

E.g.f. satisfies: A(x) = ( x / Series_Reversion( x*A'(x)^5 ) )^(1/5).

a(n) = [x^(n-1)/(n-1)!] A(x)^(5*n-4)/(5*n-4) for n>=1.

EXAMPLE

E.g.f.: A(x) = 1 + x + x^2/2! + 11*x^3/3! + 266*x^4/4! + 10326*x^5/5! +...

such that

A(x*A'(x)^5) = A'(x) = 1 + x + 11*x^2/2! + 266*x^3/3! + 10326*x^4/4! +...

To illustrate a(n) = [x^(n-1)/(n-1)!] A(x)^(5*n-4)/(5*n-4), create a table of coefficients of x^k/k!, k>=0, in A(x)^(5*n-4), n>=1, like so:

A^1 : [1,  1,   1,    11,    266,    10326,     562926, ...];

A^6 : [1,  6,  36,   276,   4086,   124476,    6058956, ...];

A^11: [1, 11, 121,  1441,  21956,   530376,   21460736, ...];

A^16: [1, 16, 256,  4256,  79376,  1891776,   66002016, ...];

A^21: [1, 21, 441,  9471, 216846,  5697426,  191016546, ...];

A^26: [1, 26, 676, 17836, 489866, 14636076,  510313076, ...];

A^31: [1, 31, 961, 30101, 968936, 32971476, 1241800356, ...]; ...

then the diagonal in the above table generates this sequence shift left:

[1/1, 6/6, 121/11, 4256/16, 216846/21, 14636076/26, 1241800356/31, ...].

SUMS OF TERM RESIDUES MODULO 2^n.

Given a(k) == 0 (mod 2^n) for k>=(8*n-6) for n>1, then it is interesting to consider the sums of the residues of all terms modulo 2^n for n>=1.

Let b(n) = Sum_{k>=0} a(k) (mod 2^n) for n>=1, then the sequence {b(n)} begins:

[4, 16, 52, 180, 388, 868, 2532, 5860, 13028, 27364, 63204, 157412, 370404, 780004, 1730276, 3630820, 7431908, 14509796, 32597732, 72967908, ...].

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+intformal(subst(A, x, x*A'^5 +x*O(x^n)))); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+intformal((1/x*serreverse(x/A^5 +x*O(x^n)))^(1/5))); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A231619, A231866, A231899, A232694, A232696.

Sequence in context: A100841 A003389 A243210 * A027019 A255955 A285051

Adjacent sequences:  A232692 A232693 A232694 * A232696 A232697 A232698

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 27 2013

STATUS

approved

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Last modified June 19 19:09 EDT 2019. Contains 324222 sequences. (Running on oeis4.)