A232694 E.g.f. A(x) satisfies: A'(x) = A(x*A'(x)^4) with A(0)=1.
1, 1, 1, 9, 177, 5601, 249681, 14569545, 1062623265, 93853717761, 9810385567329, 1192614883442889, 166310354311947345, 26308546859152889697, 4677436610087462937393, 927353710845763536487305, 203648424149429271943770945, 49245501579619466882211194625, 13045520297945193508654786790337
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + x^2/2! + 9*x^3/3! + 177*x^4/4! + 5601*x^5/5! +... such that A(x*A'(x)^4) = A'(x) = 1 + x + 9*x^2/2! + 177*x^3/3! + 5601*x^4/4! +... To illustrate a(n) = [x^(n-1)/(n-1)!] A(x)^(4*n-3)/(4*n-3), create a table of coefficients of x^k/k!, k>=0, in A(x)^(4*n-3), n>=1, like so: A^1 : [1, 1, 1, 9, 177, 5601, 249681, 14569545, ...]; A^5 : [1, 5, 25, 165, 2145, 55125, 2211225, 120873045, ...]; A^9 : [1, 9, 81, 801, 10449, 218889, 7501761, 373998465, ...]; A^13: [1, 13, 169, 2301, 35841, 731133, 21950409, 974182989, ...]; A^17: [1, 17, 289, 5049, 95217, 2102577, 60325809, 2417773881, ...]; A^21: [1, 21, 441, 9429, 211617, 5243301, 154446201, 5861076165, ...]; A^25: [1, 25, 625, 15825, 414225, 11585625, 364238625, 13752570225, ...]; A^29: [1, 29, 841, 24621, 738369, 23206989, 791747241, 30816074685, ...]; ... then the diagonal in the above table generates this sequence shift left: [1/1, 5/5, 81/9, 2301/13, 95217/17, 5243301/21, 364238625/25, 30816074685/29, ...].
Programs
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PARI
{a(n)=local(A=1+x); for(i=1, n, A=1+intformal(subst(A, x, x*A'^4 +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^4 +x*O(x^n)))^(1/4))); n!*polcoeff(A, n)} for(n=0, 25, print1(a(n), ", "))
Formula
E.g.f. satisfies: A(x) = A'(x/A(x)^4).
E.g.f. satisfies: A(x) = ( x / Series_Reversion( x*A'(x)^4 ) )^(1/4).
a(n) = [x^(n-1)/(n-1)!] A(x)^(4*n-3)/(4*n-3) for n>=1.
Comments