A317352 G.f. satisfies: A(x) = 1 + x*A(x) * ( d/dx x*A(x)^2 ).
1, 1, 5, 42, 471, 6435, 102232, 1837630, 36719439, 805716679, 19239923577, 496514053880, 13769677836500, 408449335836132, 12906850662570996, 432942515731367894, 15367227978734187567, 575544844737119275935, 22685977410186834271463, 938867118118688412116554
Offset: 0
Keywords
Examples
O.g.f.: A(x) = 1 + x + 5*x^2 + 42*x^3 + 471*x^4 + 6435*x^5 + 102232*x^6 + 1837630*x^7 + 36719439*x^8 + 805716679*x^9 + ... where A(x) = 1 + x*A(x)^3 + 2*x^2*A(x)^2*A'(x). RELATED TABLE. The table of coefficients of x^k/k! in exp( n*x*A(x)^2 ) / A(x) begins: n=1: [1, 0, -5, -158, -7779, -563924, -56177105, -7318104450, ...]; n=2: [1, 1, 0, -94, -5968, -473688, -49352768, -6601523360, ...]; n=3: [1, 2, 7, 0, -3435, -354282, -40709709, -5723430444, ...]; n=4: [1, 3, 16, 130, 0, -199016, -29893568, -4657391616, ...]; n=5: [1, 4, 27, 302, 4541, 0, -16486865, -3372747590, ...]; n=6: [1, 5, 40, 522, 10416, 251976, 0, -1833979680, ...]; n=7: [1, 6, 55, 796, 17877, 567562, 20138467, 0, ...]; n=8: [1, 7, 72, 1130, 27200, 958968, 44592256, 2176638976, 0, ...]; ... in which the coefficient of x^n in row n forms a diagonal of zeros.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Programs
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PARI
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A*(x*A^2)' +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)^2 ) * (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)^2 ) * (m+1 - m/Ser(A)))[#A]/m ); A[n+1]} for(n=0,30, print1(a(n),", "))
Formula
O.g.f. A(x) satisfies:
(1) A(x) = 1 + x*A(x) * ( d/dx x*A(x)^2 ).
(2) [x^n] exp( n * x*A(x)^2 ) / A(x) = 0 for n>0.
(3.a) [x^n] exp(-n * x*A(x)^2) * (2 - 1/A(x)) = 0 for n >= 1.
(3.b) [x^n] exp(-n^2 * x*A(x)^2) * (n + 1 - n/A(x)) = 0 for n >= 1.
(3.c) [x^n] exp(-n^(p+1) * x*A(x)^2) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0.
a(n) ~ c * 2^n * n! * n^(3/2), where c = 0.188286926603706833845600622... - Vaclav Kotesovec, Aug 05 2018