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A232696 E.g.f. A(x) satisfies: A'(x) = A(x/A'(x)) with A(0)=1. 7
1, 1, 1, -1, 2, 6, -264, 5370, -93750, 1315706, -3543736, -880688376, 56549341380, -2612825765748, 99009763750128, -2593891139126560, -31860555469490020, 12585468136754891100, -1364794494618494224128, 114095029934565534862680, -7984695190944325311086112, 419424013080533747232201968 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

CONJECTURES.

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*n-6) for k>1.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..160

FORMULA

E.g.f. satisfies: A(x) = A'(x*A(x)).

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

a(n) = [x^n/n!] A'(x)^(n+1)/(n+1) for n>=0.

EXAMPLE

E.g.f.: A(x) = 1 + x + x^2/2! - x^3/3! + 2*x^4/4! + 6*x^5/5! - 264*x^6/6! + 5370*x^7/7! +...

such that A(x) = A'(x*A(x)) and

A(x/A'(x)) = A'(x) = 1 + x - x^2/2! + 2*x^3/3! + 6*x^4/4! - 264*x^5/5! +...

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

A'^1: [1, 1, -1,   2,   6, -264,  5370,  -93750,  1315706, ...];

A'^2: [1, 2,  0,  -2,  34, -508,  7472, -100392,   774076, ...];

A'^3: [1, 3,  3,  -6,  48, -522,  6036,  -54030,  -435618, ...];

A'^4: [1, 4,  8,  -4,  36, -336,  2832,    3672, -1469680, ...];

A'^5: [1, 5, 15,  10,  10, -100,  -130,   44490, -1964390, ...];

A'^6: [1, 6, 24,  42,   6,   36, -1680,   59520, -1938564, ...];

A'^7: [1, 7, 35,  98,  84,   42, -1848,   54978, -1605394, ...];

A'^8: [1, 8, 48, 184, 328,  128, -1504,   42960, -1194368, ...];

A'^9: [1, 9, 63, 306, 846,  864, -1278,   32202,  -843750, ...]; ...

then the diagonal in the above table generates this sequence:

[1/1, 2/2, 3/3, -4/4, 10/5, 36/6, -1848/7, 42960/8, -843750/9, ...].

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, 40, 144, 432, 1008, 3184, 6384, 15600, 33520, 75504, 159472, 356080, 798448, 1797872, 3895024, 8089328, 16609008, 37842672, 76639984, 166817520, ...].

PROG

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

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

CROSSREFS

Cf. A231619, A231866, A231899, A232694, A232695.

Sequence in context: A183398 A052342 A199125 * A007190 A164829 A028337

Adjacent sequences:  A232693 A232694 A232695 * A232697 A232698 A232699

KEYWORD

sign

AUTHOR

Paul D. Hanna, Nov 28 2013

STATUS

approved

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Last modified June 4 10:51 EDT 2020. Contains 334825 sequences. (Running on oeis4.)