A319939
O.g.f. A(x) satisfies: [x^n] exp(-n^2*A(x)) / (1 - n*x)^n = 0, for n > 0.
Original entry on oeis.org
1, 1, 3, 24, 325, 6642, 176204, 5828160, 228372291, 10374419250, 534203188948, 30762752950224, 1956914341159778, 136286437739608492, 10310240639621093400, 841935232438747348480, 73807352585103519962815, 6913603998931859925828282, 689148541231545351838902508, 72838943589708142133363904400, 8137053663063956034586144506558, 958035702236154579666369909892724
Offset: 1
O.g.f.: A(x) = x + x^2 + 3*x^3 + 24*x^4 + 325*x^5 + 6642*x^6 + 176204*x^7 + 5828160*x^8 + 228372291*x^9 + 10374419250*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n^2*A(x)) / (1 - n*x)^n begins:
n=1: [1, 0, -1, -16, -567, -38816, -4771025, -886931424, ...];
n=2: [1, 0, 0, -40, -2112, -154464, -19097600, -3549131520, ...];
n=3: [1, 0, 9, 0, -3483, -333504, -43269795, -8050921776, ...];
n=4: [1, 0, 32, 224, 0, -454016, -75031040, -14515172352, ...];
n=5: [1, 0, 75, 800, 21225, 0, -92559125, -22271154000, ...];
n=6: [1, 0, 144, 1944, 88128, 2515104, 0, -25624491264, ...];
n=7: [1, 0, 245, 3920, 252693, 10516576, 505622425, 0, ...];
n=8: [1, 0, 384, 7040, 602112, 30829056, 2210682880, 134210187264, 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 + 25*x^3/3! + 673*x^4/4! + 42501*x^5/5! + 5048251*x^6/6! + 924544573*x^7/7! + 242568147585*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^2*x*Ser(A))/(1-m*x +x^2*O(x^m))^m)[m+1]/m^2 ); A[n]}
for(n=1,30,print1(a(n),", "))
A319940
O.g.f. A(x) satisfies: [x^n] exp( -n*A(x) ) / (1 - n*x)^n = 0, for n > 0.
Original entry on oeis.org
1, 3, 30, 586, 17430, 696744, 34892228, 2095250576, 146470011822, 11669877667640, 1043022527852272, 103294254944725680, 11223660850862809960, 1327297414140637610776, 169690627501555713200460, 23320015259500560303564736, 3428111061331035575475494598, 536769111685159965192282250632, 89187403511916331132476542213808
Offset: 1
O.g.f.: A(x) = x + 3*x^2 + 30*x^3 + 586*x^4 + 17430*x^5 + 696744*x^6 + 34892228*x^7 + 2095250576*x^8 + 146470011822*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x))/(1 - n*x)^n begins:
n=1: [1, 0, -5, -178, -13983, -2082676, -500286245, ...];
n=2: [1, 2, 0, -344, -30592, -4460832, -1052294144, ...];
n=3: [1, 6, 45, 0, -46323, -7614918, -1758528063, ...];
n=4: [1, 12, 184, 2960, 0, -10429504, -2724259328, ...];
n=5: [1, 20, 495, 14050, 391505, 0, -3527335025, ...];
n=6: [1, 30, 1080, 44712, 2022912, 86720544, 0, ...];
n=7: [1, 42, 2065, 115556, 7166733, 472602158, 28883187781, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 7*x^2/2! + 199*x^3/3! + 14929*x^4/4! + 2175121*x^5/5! + 516079351*x^6/6! + 179777047927*x^7/7! + ...
exp(-A(x)) = 1 - x - 5*x^2/2! - 163*x^3/3! - 13271*x^4/4! - 2012761*x^5/5! - 487790189*x^6/6! - 172048095115*x^7/7! + ...
-
{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)[m+1]/m ); A[n]}
for(n=1, 30, print1(a(n), ", "))
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! + ...
-
{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), ", "))
A320418
O.g.f. A(x) satisfies: [x^n] exp(-n^3*A(x)) / (1 - n^3*x) = 0, for n > 0.
Original entry on oeis.org
1, 4, 243, 90624, 85137125, 166983141708, 584752700234290, 3335932982893551104, 28979545952901285126801, 364345886028800419659490500, 6369639791888600743755572216796, 149926538998807813901526056378836416, 4626572216398455689960837772846170271886, 183057653659252604698726467223480475509456616, 9112803025595308606953230928489236750492759413500
Offset: 1
O.g.f.: A(x) = x + 4*x^2 + 243*x^3 + 90624*x^4 + 85137125*x^5 + 166983141708*x^6 + 584752700234290*x^7 + 3335932982893551104*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n^3*A(x)) / (1 - n^3*x) begins:
n=1: [1, 0, -7, -1456, -2174823, -10216353056, -120227612463575, ...];
n=2: [1, 0, 0, -10640, -17375232, -81730853568, -961821732684800, ...];
n=3: [1, 0, 513, 0, -54746199, -275499911232, -3246531106517055, ...];
n=4: [1, 0, 3584, 430976, 0, -612637136384, -7685860529991680, ...];
n=5: [1, 0, 14625, 3724000, 1834643625, 0, -14123406888329375, ...];
n=6: [1, 0, 44928, 19840464, 18646474752, 17991609015744, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 9*x^2/2! + 1483*x^3/3! + 2181049*x^4/4! + 10227462141*x^5/5! + 120289476378841*x^6/6! + 2947997248178316559*x^7/7! + ...
exp(-A(x)) = 1 - x - 7*x^2/2! - 1435*x^3/3! - 2168999*x^4/4! - 10205478941*x^5/5! - 120166314345239*x^6/6! - 2946310403245714303*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^3*x*Ser(A))/(1-m^3*x +x^2*O(x^m)))[m+1]/m^3 ); A[n]}
for(n=1, 20, print1(a(n), ", "))
A321085
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, 36, 330, 4092, 63308, 1165952, 24802704, 596862420, 16003332092, 472588688880, 15231153085304, 531801227022912, 19992702649780800, 805099660439308672, 34574148236596003008, 1577229602738525889984, 76172611622917782581040, 3882814240467405224924960, 208333046141920439476582608, 11737307286279509352537747760
Offset: 1
O.g.f.: A(x) = x + 2*x^2 + 6*x^3 + 36*x^4 + 330*x^5 + 4092*x^6 + 63308*x^7 + 1165952*x^8 + 24802704*x^9 + 596862420*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, -819, -39056, -2923925, -317422764, ...];
n=2: [1, 0, 0, -32, -1392, -75552, -5832320, -635767680, ...];
n=3: [1, 0, 3, 0, -1323, -100008, -8542665, -955410984, ...];
n=4: [1, 0, 8, 80, 0, -89024, -10215680, -1248268032, ...];
n=5: [1, 0, 15, 220, 3405, 0, -8752325, -1409888100, ...];
n=6: [1, 0, 24, 432, 9936, 234144, 0, -1176833664, ...];
n=7: [1, 0, 35, 728, 20853, 710248, 23232055, 0, ...];
n=8: [1, 0, 48, 1120, 37632, 1560192, 72348160, 3135469056, 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! + 49*x^3/3! + 1081*x^4/4! + 46001*x^5/5! + 3272701*x^6/6! + 345526945*x^7/7! + 50126588849*x^8/8! + ...
exp(-A(x)) = 1 - x - 3*x^2/2! - 25*x^3/3! - 695*x^4/4! - 34401*x^5/5! - 2665019*x^6/6! - 295314937*x^7/7! - 44140455855*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 1-5 of 5 results.
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