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.

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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

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), ", "))
Showing 1-2 of 2 results.

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