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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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A304861 O.g.f. A(x) satisfies: 0 = [x^n] exp( n*(n-1) * Integral 1/A(x) dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 0, 2, 20, 328, 7664, 231744, 8560512, 372339840, 18593869184, 1046764673152, 65518908623360, 4510397034460160, 338534873778165760, 27505042556295458816, 2404499023598887772160, 225014884122460397678592, 22441327480906466274779136, 2376060993772932821157273600, 266169866452350363506325897216, 31451236460722731478509841711104
Offset: 0

Views

Author

Paul D. Hanna, Jun 01 2018

Keywords

Comments

Note: 0 = [x^n] exp( (n-1) * Integral 1/F(x) dx ) / F(x) holds for n > 0 when F(x) = sqrt(1 + x^2).
Note: 0 = [x^n] exp( n * Integral 1/G(x) dx ) / G(x) holds for n > 0 when G(x) = 1 + x.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + 2*x^2 + 20*x^3 + 328*x^4 + 7664*x^5 + 231744*x^6 + 8560512*x^7 + 372339840*x^8 + 18593869184*x^9 + 1046764673152*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in  exp(n*(n-1) * Integral 1/A(x) dx)/A(x) begins:
n=0: [1, 0, -2, -20, -324, -7584, -230040, -8516976, ...];
n=1: [1, 0, -2, -20, -324, -7584, -230040, -8516976, ...];
n=2: [1, 2, 0, -24, -380, -8424, -248640, -9062720, ...];
n=3: [1, 6, 16, 0, -480, -10528, -292544, -10293696, ...];
n=4: [1, 12, 70, 236, 0, -13472, -378336, -12576960, ...];
n=5: [1, 20, 198, 1260, 5176, 0, -485520, -16616864, ...];
n=6: [1, 30, 448, 4400, 31176, 151792, 0, -21316608, ...];
n=7: [1, 42, 880, 12216, 125340, 989384, 5588416, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp( n*(n-1) * Integral 1/A(x) dx ) / A(x), for n > 0.
RELATED SERIES.
1/A(x) = 1 - 2*x^2 - 20*x^3 - 324*x^4 - 7584*x^5 - 230040*x^6 - 8516976*x^7 - 371005040*x^8 - 18545507840*x^9 - 1044727771680*x^10 + ...
exp(Integral 1/A(x) dx) = 1 + 2*x/2 + 2*x^2/2^2 - 4*x^3/2^3 - 90*x^4/2^4 - 2244*x^5/2^5 - 85196*x^6/2^6 - 4372040*x^7/2^7 - 281105594*x^8/2^8 - 21659046420*x^9/2^9 + ...
exp(2 * Integral 1/A(x) dx) = 1 + 2*x + 2*x^2 - 12*x^4 - 152*x^5 - 2808*x^6 - 71040*x^7 - 2265680*x^8 - 86833824*x^9 - 3878209440*x^10 - 197532405760*x^11 + ..., an integer series.
A'(x)/A(x) = 4*x + 60*x^2 + 1304*x^3 + 38120*x^4 + 1385344*x^5 + 59770928*x^6 + 2973371104*x^7 + 167126930016*x^8 + 10457452841984*x^9 + ...

Links

Crossrefs

Cf. A304862, A305144, A305145, A305146, A305147.
Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.

Programs

  • PARI
    {a(n) = my(A=[1],m); for(i=1,n+1, m=#A; A=concat(A,0); A[m+1] = Vec( exp(m*(m-1)*intformal(1/Ser(A))) / Ser(A) )[m+1] );A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

a(n) ~ sqrt(1-c) * 2^(2*n - 2) * n^(n - 1/2) / (sqrt(Pi) * exp(n) * c^(n - 1/2) * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, Oct 18 2020

A304862 O.g.f. A(x) satisfies: 0 = [x^n] exp( n*(n+1) * Integral 1/A(x) dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 2, 4, 32, 512, 12000, 366400, 13688960, 602193152, 30397531136, 1728411805184, 109177081065472, 7578667350118400, 573143826340921344, 46886796648225349632, 4124437046595970498560, 388153835886455237115904, 38910750374376922179960832, 4139100381105952654252048384, 465644313330130076144183017472
Offset: 0

Views

Author

Paul D. Hanna, Jun 01 2018

Keywords

Comments

Note: 0 = [x^n] exp( n * Integral 1/G(x) dx ) / G(x) holds for n > 0 when G(x) = 1 + x.
Note: 0 = [x^n] exp( (n-1) * Integral 1/F(x) dx ) / F(x) holds for n > 0 when F(x) = sqrt(1 + x^2).
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + 2*x + 4*x^2 + 32*x^3 + 512*x^4 + 12000*x^5 + 366400*x^6 + 13688960*x^7 + 602193152*x^8 + 30397531136*x^9 + 1728411805184*x^10 + 109177081065472*x^11 + 7578667350118400*x^12 + 573143826340921344*x^13 + 46886796648225349632*x^14 + 4124437046595970498560*x^15 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in  exp(n*(n+1) * Integral 1/A(x) dx)/A(x) begins:
n=0: [1, -2, 0, -24, -400, -10080, -319872, -12251008, ...];
n=1: [1, 0, -4, -80/3, -456, -165536/15, -3089536/9, ...];
n=2: [1, 4, 0, -48, -616, -66816/5, -1985184/5, ...];
n=3: [1, 10, 36, 0, -976, -93312/5, -500928, ...];
n=4: [1, 18, 140, 1648/3, 0, -83680/3, -6379648/9, ...];
n=5: [1, 28, 360, 2736, 12200, 0, -1023072, ...];
n=6: [1, 40, 756, 8880, 70664, 1800288/5, 0,  ...];
n=7: [1, 54, 1400, 69256/3, 269184, 34495552/15, 599302144/45, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp( n*(n+1) * Integral 1/A(x) dx ) / A(x), for n > 0.
RELATED SERIES.
1/A(x) = 1 - 2*x - 24*x^3 - 400*x^4 - 10080*x^5 - 319872*x^6 - 12251008*x^7 - 548218368*x^8 - 28018713600*x^9 - 1608234580480*x^10 + ...
exp(Integral 1/A(x) dx) = 1 + x - x^2/2! - 5*x^3/3! - 143*x^4/4! - 10279*x^5/5! - 1265009*x^6/6! - 238548701*x^7/7! - 63550271455*x^8/8! - 22650892439183*x^9/9! + ...
A'(x)/A(x) = 2 + 4*x + 80*x^2 + 1808*x^3 + 54912*x^4 + 2052736*x^5 + 90617984*x^6 + 4595611904*x^7 + 262620131840*x^8 + 16670924217344*x^9 + ...

Links

Crossrefs

Cf. A304861, A305144, A305145, A305146, A305147.
Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.

Programs

  • PARI
    {a(n) = my(A=[1],m); for(i=1,n+1, m=#A; A=concat(A,0); A[m+1] = Vec( exp(m*(m+1)*intformal(1/Ser(A))) / Ser(A) )[m+1] );A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 0.08310334422... - Vaclav Kotesovec, Oct 19 2020
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