A319938
O.g.f. A(x) satisfies: [x^n] exp(-n*A(x)) / (1 - n*x) = 0, for n > 0.
Original entry on oeis.org
1, 1, 3, 18, 165, 2019, 30688, 554784, 11591649, 274313325, 7242994143, 210931834662, 6713206636084, 231754182524900, 8624280230971980, 344124280164153056, 14656294893872323449, 663624782214112471329, 31833832291287920426617, 1612762327644980719082470, 86050799297228500838101677, 4823357354919905244973170883, 283375597845431500054861239512
Offset: 1
O.g.f.: A(x) = x + x^2 + 3*x^3 + 18*x^4 + 165*x^5 + 2019*x^6 + 30688*x^7 + 554784*x^8 + 11591649*x^9 + 274313325*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) / (1 - n*x) begins:
n=1: [1, 0, -1, -16, -423, -19616, -1444625, -154014624, ...];
n=2: [1, 0, 0, -20, -768, -38832, -2895680, -308705280, ...];
n=3: [1, 0, 3, 0, -783, -53568, -4309605, -465802704, ...];
n=4: [1, 0, 8, 56, 0, -50144, -5307200, -616050432, ...];
n=5: [1, 0, 15, 160, 2265, 0, -4729025, -711963600, ...];
n=6: [1, 0, 24, 324, 6912, 145584, 0, -613885824, ...];
n=7: [1, 0, 35, 560, 15057, 460768, 13696795, 0, ...];
n=8: [1, 0, 48, 880, 28032, 1050432, 44437120, 1769051136, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 529*x^4/4! + 22581*x^5/5! + 1598011*x^6/6! + 166508413*x^7/7! + 23765885025*x^8/8! + ...
exp(-A(x)) = 1 - x - x^2/2! - 13*x^3/3! - 359*x^4/4! - 17501*x^5/5! - 1326929*x^6/6! - 143902249*x^7/7! - 21072159247*x^8/8! + ...
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{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp(-m*x*Ser(A))/(1-m*x +x^2*O(x^m)))[m+1]/m ); A[n]}
for(n=1,30,print1(a(n),", "))
A321086
O.g.f. A(x) satisfies: [x^n] exp(n*A(x)) * (1 - n*x - n*x^2) = 0, for n > 0.
Original entry on oeis.org
1, 2, 6, 32, 220, 1812, 17108, 180512, 2093760, 26396160, 358741328, 5223336288, 81079811280, 1336407320080, 23311138957200, 429063111959808, 8311760620707648, 169072470759431232, 3603666131945918144, 80327823251439861760, 1869212211081119135616, 45331401566332423284864, 1143967734536203174726784, 29996686272924492809481216, 816185909551276017516640000
Offset: 1
O.g.f.: A(x) = x + 2*x^2 + 6*x^3 + 32*x^4 + 220*x^5 + 1812*x^6 + 17108*x^7 + 180512*x^8 + 2093760*x^9 + 26396160*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) * (1 - n*x - n*x^2) begins:
n=1: [1, 0, 1, 28, 729, 26416, 1321225, 87466716, ...];
n=2: [1, 0, 0, 32, 1200, 49152, 2569600, 172974720, ...];
n=3: [1, 0, -3, 0, 1089, 60408, 3509325, 246760776, ...];
n=4: [1, 0, -8, -80, 0, 49024, 3777280, 293683968, ...];
n=5: [1, 0, -15, -220, -2535, 0, 2848825, 291386100, ...];
n=6: [1, 0, -24, -432, -7056, -105984, 0, 208089216, ...];
n=7: [1, 0, -35, -728, -14175, -293048, -5733875, 0, ...];
n=8: [1, 0, -48, -1120, -24576, -590592, -15603200, -391709184, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
(a) Differential Equation.
O.g.f. A(x) satisfies: A(x) = x + x^2 + x*A(x)*A'(x) where
A'(x) = 1 + 4*x + 18*x^2 + 128*x^3 + 1100*x^4 + 10872*x^5 + 119756*x^6 + ...
A(x)*A'(x) = x + 6*x^2 + 32*x^3 + 220*x^4 + 1812*x^5 + 17108*x^6 + 17108*x^7 + ...
so that A(x) - x*A(x)*A'(x) = x + x^2.
(b) Exponentiation.
exp(A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 985*x^4/4! + 32321*x^5/5! + 1544701*x^6/6! + 99637105*x^7/7! + 8257877489*x^8/8! + ...
exp(-A(x)) = 1 - x - 3*x^2/2! - 25*x^3/3! - 599*x^4/4! - 21681*x^5/5! - 1106939*x^6/6! - 74873737*x^7/7! - 6431021295*x^8/8! + ...
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{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = -Vec( exp(m*x*Ser(A))*(1-m*x-m*x^2 +x^2*O(x^m)))[m+1]/m ); A[n]}
for(n=1, 30, print1(a(n), ", "))
Showing 1-2 of 2 results.
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