A317346
O.g.f. A(x) satisfies: [x^n] exp( n^3*x - n^2*A(x) ) = 0 for n >= 1.
Original entry on oeis.org
1, 2, 72, 8096, 1839000, 695334816, 392764566208, 309340607492096, 323795915817507936, 434750954619876448000, 728547799352068864173632, 1490865523016798790557180928, 3659466509860384349989504297344, 10614823215131644149237135937187328, 35927108634064565449228268842108588800, 140351379904337650357154561973550135705600
Offset: 1
O.g.f.: A(x) = x + 2*x^2 + 72*x^3 + 8096*x^4 + 1839000*x^5 + 695334816*x^6 + 392764566208*x^7 + 309340607492096*x^8 + ...
such that [x^n] exp( n^3*x - n^2*A(x) ) = 0 for n >= 1.
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^3*x - n^2*A(x) ) begins:
n=1: [1, 0, -4, -432, -194256, -220662720, -500627544000, ...];
n=2: [1, 4, 0, -1856, -805376, -898258176, -2023715201024, ...];
n=3: [1, 18, 288, 0, -1989792, -2154563712, -4727980751616, ...];
n=4: [1, 48, 2240, 94464, 0, -4244861952, -9137589559296, ...];
n=5: [1, 100, 9900, 959200, 84852400, 0, -15901448888000, ...];
n=6: [1, 180, 32256, 5738688, 1003636224, 161358324480, 0, ...];
n=7: [1, 294, 86240, 25218144, 7335234144, 2103824749824, 557359956846336, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 5*x^2/2! + 445*x^3/3! + 196105*x^4/4! + 221673401*x^5/5! + 501981700621*x^6/6! + 1983064113021685*x^7/7! + ... + A317345(n)*x^n/n! + ...
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{a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*x +x*O(x^#A)) / Ser(A)^(m^2) )[m+1]/m^2 ); polcoeff( log(Ser(A)),n)}
for(n=1,20,print1(a(n),", "))
A319941
O.g.f. A(x) satisfies: [x^n] exp( n^4*x - n*A(x) ) = 0 for n >= 1.
Original entry on oeis.org
1, 49, 22542, 34776266, 124857847020, 863035137487572, 10208133235178252640, 190511518719216943969008, 5284939084238999180631562560, 208156037245304153601560603185040, 11224507767787823723649649410800624768, 804502870984274832989329177960786158548256
Offset: 1
G.f.: A(x) = x + 49*x^2 + 22542*x^3 + 34776266*x^4 + 124857847020*x^5 + 863035137487572*x^6 + 10208133235178252640*x^7 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^4*x - n*A(x) ) begins:
n=1: [1, 0, -98, -135252, -834601572, -14982809095440, ...];
n=2: [1, 14, 0, -275992, -1684485824, -30082728311616, ...];
n=3: [1, 78, 5790, 0, -2603944836, -45947242627272, ...];
n=4: [1, 252, 63112, 15165648, 0, -63525640595328, ...];
n=5: [1, 620, 383910, 236740340, 140783667580, 0, ...];
n=6: [1, 1290, 1663512, 2143601928, 2754163718208, 3423991878509760, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 99*x^2/2! + 135547*x^3/3! + 835200793*x^4/4! + 14987248838841*x^5/5! + 621476619810599851*x^6/6! + ...
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{a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^4*x +x*O(x^#A)) / Ser(A)^m )[m+1]/m ); polcoeff( log(Ser(A)), n)}
for(n=1, 15, print1(a(n), ", "))
A319942
O.g.f. A(x) satisfies: [x^n] exp( n^4*x - n^2*A(x) ) = 0 for n >= 1.
Original entry on oeis.org
1, 18, 5616, 6776352, 20200266000, 119799079486272, 1242032508354758400, 20634722510624457007104, 515736410631216295520236032, 18480208067078637967802351884800, 913939154183946975187574927409795072, 60487962958244860971401604975128195088384
Offset: 1
G.f.: A(x) = x + 18*x^2 + 5616*x^3 + 6776352*x^4 + 20200266000*x^5 + 119799079486272*x^6 + 1242032508354758400*x^7 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^4*x - n^2*A(x) ) begins:
n=1: [1, 0, -36, -33696, -162628560, -2424019789440, ...];
n=2: [1, 12, 0, -138240, -657040896, -9735157974528, ...];
n=3: [1, 72, 4860, 0, -1533920976, -22357116073728, ...];
n=4: [1, 240, 57024, 12870144, 0, -41496660080640, ...];
n=5: [1, 600, 359100, 213537600, 121570858800, 0, ...];
n=6: [1, 1260, 1586304, 1994264064, 2496165050880, 3006510865205760, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 37*x^2/2! + 33805*x^3/3! + 162771337*x^4/4! + 2424857569561*x^5/5! + 86269983111064621*x^6/6! + 6260449705448367386917*x^7/7! + ...
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{a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^4*x +x*O(x^#A)) / Ser(A)^(m^2) )[m+1]/m^2 ); polcoeff( log(Ser(A)), n)}
for(n=1, 15, print1(a(n), ", "))
Showing 1-3 of 3 results.
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