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 1-3 of 3 results.

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

Views

Author

Paul D. Hanna, Oct 09 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 + 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! + ...

Links

Crossrefs

Cf. A319938, A319940, A320669.

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*x +x^2*O(x^m))^m)[m+1]/m^2 ); A[n]}
for(n=1,30,print1(a(n),", "))

Formula

a(n) ~ c * d^n * n! / n^3, where d = 6.1103392... and c = 0.05165... - Vaclav Kotesovec, Oct 24 2020

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), ", "))

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 1-3 of 3 results.

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