A320417
O.g.f. A(x) satisfies: [x^n] exp(-n^2*A(x)) / (1 - n^2*x) = 0, for n > 0.
Original entry on oeis.org
1, 2, 27, 1312, 125725, 19877634, 4644661441, 1501087818944, 640786440035745, 349236672544961550, 236695639072681655042, 195322914258394193939808, 192869728403705883411146031, 224593016480452799339762161070, 304623945406240486265488269648600, 476130992607087098886173799883802624, 849656108159062192953462986972010725625
Offset: 1
O.g.f.: A(x) = x + 2*x^2 + 27*x^3 + 1312*x^4 + 125725*x^5 + 19877634*x^6 + 4644661441*x^7 + 1501087818944*x^8 + 640786440035745*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n^2*A(x)) / (1 - n^2*x) begins:
n=1: [1, 0, -3, -160, -31455, -15082176, -14310224075, ...];
n=2: [1, 0, 0, -520, -124416, -60323424, -57244390400, ...];
n=3: [1, 0, 45, 0, -237951, -134365824, -128906646075, ...];
n=4: [1, 0, 192, 5600, 0, -205474176, -226875814400, ...];
n=5: [1, 0, 525, 27200, 2383425, 0, -306673758875, ...];
n=6: [1, 0, 1152, 87480, 12925440, 1915825824, 0, ...];
n=7: [1, 0, 2205, 227360, 47631969, 11053430976, 2730401653525, 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! + 175*x^3/3! + 32209*x^4/4! + 15252821*x^5/5! + 14405086381*x^6/6! + 23511056196475*x^7/7! + ...
exp(-A(x)) = 1 - x - 3*x^2/2! - 151*x^3/3! - 30815*x^4/4! - 14924901*x^5/5! - 14219731019*x^6/6! - 23307795465907*x^7/7! + ...
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{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(-m^2*x*Ser(A))/(1-m^2*x +x^2*O(x^m))^1)[m+1]/m^2 ); A[n]}
for(n=1, 30, print1(a(n), ", "))
A320669
O.g.f. A(x) satisfies: [x^n] exp(-n^3*A(x)) / (1 - n^2*x)^n = 0, for n > 0.
Original entry on oeis.org
1, 2, 27, 2176, 316125, 92433420, 38689900249, 24036220587520, 19705732103751309, 21228545767337495500, 28631298365231328948940, 47701162183511368703635200, 95797470923250302955913961043, 228907109818475997814838969598324, 641132565508623116202107427900402750, 2082400670957118326405938988144017645568
Offset: 1
O.g.f.: A(x) = x + 2*x^2 + 27*x^3 + 2176*x^4 + 316125*x^5 + 92433420*x^6 + 38689900249*x^7 + 24036220587520*x^8 + 19705732103751309*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n^3*A(x)) / (1 - n^2*x)^n begins:
n=1: [1, 0, -3, -160, -52191, -37930176, -66549456875, ...];
n=2: [1, 0, 0, -1040, -414720, -303430848, -532404700160, ...];
n=3: [1, 0, 135, 0, -1237275, -1019993472, -1799293659165, ...];
n=4: [1, 0, 768, 22400, 0, -2155144704, -4259850874880, ...];
n=5: [1, 0, 2625, 136000, 25862625, 0, -7511859284375, ...];
n=6: [1, 0, 6912, 524880, 192513024, 36792874944, 0, ...];
n=7: [1, 0, 15435, 1591520, 938926485, 280095248832, 121196964253015, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
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{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(-m^3*x*Ser(A))/(1-m^2*x +x^2*O(x^m))^m)[m+1]/m^3 ); A[n]}
for(n=1, 20, print1(a(n), ", "))
A320668
O.g.f. A(x) satisfies: [x^n] exp(-n^3*A(x)) / (1 - n*x)^(n^2) = 0, for n > 0.
Original entry on oeis.org
1, 1, 3, 48, 1125, 74844, 4538576, 571979264, 61768818081, 11756208796500, 1930305045364047, 501690433505046336, 114627985830970025544, 38401761759325497631504, 11530876917646339177773375, 4792821920208552461683208192, 1816651428077402993910096849969, 911361374568809242258003199407404
Offset: 1
O.g.f.: A(x) = x + x^2 + 3*x^3 + 48*x^4 + 1125*x^5 + 74844*x^6 + 4538576*x^7 + 571979264*x^8 + 61768818081*x^9 + 11756208796500*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n^3*A(x)) / (1 - n*x)^(n^2) begins:
n=1: [1, 0, -1, -16, -1143, -134816, -53867825, ...];
n=2: [1, 0, 0, -80, -8832, -1076928, -431006720, ...];
n=3: [1, 0, 27, 0, -24543, -3592512, -1464710445, ...];
n=4: [1, 0, 128, 896, 0, -7099904, -3495833600, ...];
n=5: [1, 0, 375, 4000, 371625, 0, -6020725625, ...];
n=6: [1, 0, 864, 11664, 2270592, 78335424, 0, ...];
n=7: [1, 0, 1715, 27440, 9134433, 444056032, 73395100555, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
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{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(-m^3*x*Ser(A))/(1-m*x +x^2*O(x^m))^(m^2))[m+1]/m^3 ); A[n]}
for(n=1, 20, print1(a(n), ", "))
Showing 1-3 of 3 results.
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