cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Paul D. Hanna, Oct 09 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.
Compare to: [x^n] exp(-n*G(x)) * (1 + n*x) = 0, for n > 0, when G(x) = x - x*G(x)*G'(x), where G(-x)/(-x) is the o.g.f. of A088716.

Examples

			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! + ...

Links

Crossrefs

Cf. A088716, A321085, A319939, A319940, A320417.

Programs

  • PARI
    {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),", "))

Formula

a(n) ~ c * n^(n-1), where c = 0.335949071234... - Vaclav Kotesovec, Oct 22 2020

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

Views

Author

Paul D. Hanna, Oct 11 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.
a(n) is even for n > 2, with a(2^k + 1) = 2 (mod 4) for k >= 1 (conjecture).

Examples

			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! + ...

Links

Crossrefs

Cf. A319938, A319939.

Programs

  • PARI
    {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), ", "))

Formula

a(n) ~ c * d^n * n! / n^2, where d = 9.669628447... and c = 0.0559981... - Vaclav Kotesovec, Oct 24 2020

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

Views

Author

Paul D. Hanna, Oct 15 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.

Examples

			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! + ...

Links

Crossrefs

Cf. A319938, A319939, A320418.

Programs

  • PARI
    {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), ", "))

Formula

a(n) ~ c * n^(2*n - 2), where c = exp(exp(-2) - 3) * (exp(2) - 1) = 0.36419050799963000048040121372730789359398... - Vaclav Kotesovec, Aug 11 2021, updated Mar 18 2024

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

Views

Author

Paul D. Hanna, Oct 15 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.

Examples

			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! + ...

Crossrefs

Cf. A320417, A319938, A319939, A320668, A320669.

Programs

  • PARI
    {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), ", "))

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

Views

Author

Paul D. Hanna, Oct 19 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.

Examples

			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.

Crossrefs

Cf. A320418, A320668, A319939.

Programs

  • PARI
    {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

Views

Author

Paul D. Hanna, Oct 19 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.

Examples

			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.

Crossrefs

Cf. A320418, A320669, A319939.

Programs

  • PARI
    {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 1-6 of 6 results.

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