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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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

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
Showing 1-2 of 2 results.

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