cp's OEIS Frontend

A232696 E.g.f. A(x) satisfies: A'(x) = A(x/A'(x)) with A(0)=1.

%I A232696 #11 Dec 03 2013 20:51:28
%S A232696 1,1,1,-1,2,6,-264,5370,-93750,1315706,-3543736,-880688376,
%T A232696 56549341380,-2612825765748,99009763750128,-2593891139126560,
%U A232696 -31860555469490020,12585468136754891100,-1364794494618494224128,114095029934565534862680,-7984695190944325311086112,419424013080533747232201968
%N A232696 E.g.f. A(x) satisfies: A'(x) = A(x/A'(x)) with A(0)=1.
%C A232696  CONJECTURES.
%C A232696 a(n) == 0 (mod 2) for n>=4.
%C A232696 a(n) == 0 (mod 2^2) for n>=10.
%C A232696 a(n) == 0 (mod 2^3) for n>=18.
%C A232696 a(n) == 0 (mod 2^k) for n>=(8*n-6) for k>1.
%H A232696 Paul D. Hanna, <a href="/A232696/b232696.txt">Table of n, a(n) for n = 0..160</a>
%F A232696 E.g.f. satisfies: A(x) = A'(x*A(x)).
%F A232696 E.g.f. satisfies: A(x) = 1/x * Series_Reversion( x/A'(x) ).
%F A232696 a(n) = [x^n/n!] A'(x)^(n+1)/(n+1) for n>=0.
%e A232696 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! +...
%e A232696 such that A(x) = A'(x*A(x)) and
%e A232696 A(x/A'(x)) = A'(x) = 1 + x - x^2/2! + 2*x^3/3! + 6*x^4/4! - 264*x^5/5! +...
%e A232696 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:
%e A232696 A'^1: [1, 1, -1,   2,   6, -264,  5370,  -93750,  1315706, ...];
%e A232696 A'^2: [1, 2,  0,  -2,  34, -508,  7472, -100392,   774076, ...];
%e A232696 A'^3: [1, 3,  3,  -6,  48, -522,  6036,  -54030,  -435618, ...];
%e A232696 A'^4: [1, 4,  8,  -4,  36, -336,  2832,    3672, -1469680, ...];
%e A232696 A'^5: [1, 5, 15,  10,  10, -100,  -130,   44490, -1964390, ...];
%e A232696 A'^6: [1, 6, 24,  42,   6,   36, -1680,   59520, -1938564, ...];
%e A232696 A'^7: [1, 7, 35,  98,  84,   42, -1848,   54978, -1605394, ...];
%e A232696 A'^8: [1, 8, 48, 184, 328,  128, -1504,   42960, -1194368, ...];
%e A232696 A'^9: [1, 9, 63, 306, 846,  864, -1278,   32202,  -843750, ...]; ...
%e A232696 then the diagonal in the above table generates this sequence:
%e A232696 [1/1, 2/2, 3/3, -4/4, 10/5, 36/6, -1848/7, 42960/8, -843750/9, ...].
%e A232696 SUMS OF TERM RESIDUES MODULO 2^n.
%e A232696 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.
%e A232696 Let b(n) = Sum_{k>=0} a(k) (mod 2^n) for n>=1, then the sequence {b(n)} begins:
%e A232696 [4, 16, 40, 144, 432, 1008, 3184, 6384, 15600, 33520, 75504, 159472, 356080, 798448, 1797872, 3895024, 8089328, 16609008, 37842672, 76639984, 166817520, ...].
%o A232696 (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)}
%o A232696 for(n=0, 25, print1(a(n), ", "))
%o A232696 (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)}
%o A232696 for(n=0, 25, print1(a(n), ", "))
%Y A232696 Cf. A231619, A231866, A231899, A232694, A232695.
%K A232696 sign
%O A232696 0,5
%A A232696 _Paul D. Hanna_, Nov 28 2013