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 1-5 of 5 results.

A305144 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral 1/A(x) dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 3, 26, 417, 9726, 295000, 10946172, 478392123, 24001955894, 1357178076996, 85294057678956, 5893597893045486, 443851259961124476, 36172543480754645712, 3171024571792211972824, 297496306299698019850371, 29738036578363255676373606, 3155172706300699135457477884, 354114794234668864071564974988, 41914947879716810639378379595146
Offset: 0

Views

Author

Paul D. Hanna, May 31 2018

Keywords

Comments

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

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 26*x^3 + 417*x^4 + 9726*x^5 + 295000*x^6 + 10946172*x^7 + 478392123*x^8 + 24001955894*x^9 + 1357178076996*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^2*Integral 1/A(x) dx)/A(x) begins:
n=0: [1, -1, -2, -21, -364, -8830, -273972, -10313037, -455135384, ...];
n=1: [1, 0, -3, -24, -390, -9264, -284235, -10625424, -466720254, ...];
n=2: [1, 3, 0, -35, -495, -10773, -318192, -11635020, -503631630, ...];
n=3: [1, 8, 25, 0, -700, -14272, -388269, -13599240, -573208625, ...];
n=4: [1, 15, 102, 371, 0, -19746, -525980, -17134953, -691326666, ...];
n=5: [1, 24, 273, 1904, 8136, 0, -716177, -23528472, -891395739, ...];
n=6: [1, 35, 592, 6381, 47945, 238403, 0, -31651620, -1235181962, ...];
n=7: [1, 48, 1125, 17080, 187110, 1536336, 8774025, 0, -1646095140, ...];
n=8: [1, 63, 1950, 39435, 583620, 6681714, 60092844, 389166915, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2 * Integral 1/A(x) dx)/A(x), for n > 0.
RELATED SERIES.
1/A(x) = 1 - x - 2*x^2 - 21*x^3 - 364*x^4 - 8830*x^5 - 273972*x^6 - 10313037*x^7 - 455135384*x^8 - 22995056286*x^9 - 1307053358940*x^10 + ...
exp( Integral 1/A(x) dx) = 1 + x - x^3 - 6*x^4 - 78*x^5 - 1544*x^6 - 40605*x^7 - 1328178*x^8 - 51857806*x^9 - 2350025232*x^10 - 121120896906*x^11 - 6991877399100*x^12 + ..., which is an integer series.
A'(x)/A(x) = 1 + 5*x + 70*x^2 + 1557*x^3 + 46316*x^4 + 1705382*x^5 + 74365572*x^6 + 3732699789*x^7 + 211429236472*x^8 + 13318438851990*x^9 + 922595879008860*x^10 + ...

Links

Crossrefs

Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.
Cf. A305145, A305146, A305147, A304861, A304862.

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^2*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 = 0.0604992010464118... - Vaclav Kotesovec, Oct 19 2020

A305145 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral 1/A(x) dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 21, 1886, 381735, 134584434, 72514796422, 55192152857400, 56287911330435339, 74043167807482274450, 122040226074154110294114, 246341047594913378800486668, 597752265070243363135031803950, 1716967839431601765698468898047292, 5762431350664488199395983555754160140, 22346478647255335081326815815314403748524
Offset: 0

Views

Author

Paul D. Hanna, May 31 2018

Keywords

Comments

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

Examples

			O.g.f.: A(x) = 1 + x + 21*x^2 + 1886*x^3 + 381735*x^4 + 134584434*x^5 + 72514796422*x^6 + 55192152857400*x^7 + 56287911330435339*x^8 + 74043167807482274450*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^3*Integral 1/A(x) dx)/A(x) begins:
n=0: [1, -1, -20, -1845, -377584, -133748650, -72227419704, -55040493806445, ...];
n=1: [1, 0, -21, -1872, -379890, -134201604, -72383437035, -55123034324112, ...];
n=2: [1, 7, 0, -2033, -396970, -137452068, -73490534208, -55705843833995, ...];
n=3: [1, 26, 304, 0, -437155, -147006370, -76635381186, -57333497856168, ...];
n=4: [1, 63, 1932, 36075, 0, -163035066, -83375170872, -60709861617885, ...];
n=5: [1, 124, 7605, 304780, 8444291, 0, -92858506104, -66905102463320, ...];
n=6: [1, 215, 22984, 1625463, 84879650, 3287781224, 0, -74725745263095, ...];
n=7: [1, 342, 58290, 6597132, 556856100, 37129859844, 1920530286186, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3 * Integral 1/A(x) dx)/A(x), for n > 0.
RELATED SERIES.
1/A(x) = 1 - x - 20*x^2 - 1845*x^3 - 377584*x^4 - 133748650*x^5 - 72227419704*x^6 - 55040493806445*x^7 - 56174066916766400*x^8 - 73928074251625193826*x^9 + ...
exp( Integral 1/A(x) dx) = 1 + x - 7*x^3 - 468*x^4 - 75978*x^5 - 22366934*x^6 - 10340491005*x^7 - 6890379290514*x^8 - 6248442860989378*x^9 - 7399048902607246248*x^10 + ..., which is an integer series.
A'(x)/A(x) = 1 + 41*x + 5596*x^2 + 1518597*x^3 + 670826996*x^4 + 434225271374*x^5 + 385813724342292*x^6 + 449847594913097949*x^7 + 665870324595294969196*x^8 + ...

Links

Crossrefs

Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.
Cf. A305144, A305146, A305147, A304861, A304862.

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^3*intformal(1/Ser(A))) / Ser(A) )[m+1] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) ~ sqrt(1-c) * 3^(3*n - 1) * n^(2*n - 1/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n - 1) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - Vaclav Kotesovec, Oct 19 2020

A305147 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral 1/A(x)^2 dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 1, 13, 201, 4799, 146509, 5465853, 239779725, 12065090215, 683788505469, 43055465865105, 2979786144976833, 224718173520876855, 18335712354871184749, 1609062791960716840469, 151097465043129176493237, 15116317905498147638860983, 1605008879121294393641990077, 180254723532204767389702764585, 21348717445490413966641543430233
Offset: 0

Views

Author

Paul D. Hanna, May 31 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.
Note: 0 = [x^n] exp( n * Integral 1/F(x)^2 dx ) / F(x) holds for n > 0 when 1/F(-x) = C(x) = 1 + x*C(x)^2 is a g.f. of A000108.

Examples

			O.g.f.: A(x) = 1 + x + x^2 + 13*x^3 + 201*x^4 + 4799*x^5 + 146509*x^6 + 5465853*x^7 + 239779725*x^8 + 12065090215*x^9 + 683788505469*x^10 + 43055465865105*x^11 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^2*Integral 1/A(x)^2 dx)/A(x) begins:
n=0: [1, -1, 0, -12, -176, -4410, -136968, -5173266, ...];
n=1: [1, 0, -3/2, -12, -1545/8, -23229/5, -2282987/16, -187096983/35, ...];
n=2: [1, 3, 0, -20, -252, -27426/5, -808448/5, -41341014/7, ...];
n=3: [1, 8, 45/2, 0, -3241/8, -37566/5, -16103943/80, -98105421/14, ...];
n=4: [1, 15, 96, 308, 0, -57474/5, -282824, -315815478/35, ...];
n=5: [1, 24, 525/2, 1688, 50967/8, 0, -6694523/16, -179820699/14, ...];
n=6: [1, 35, 576, 5868, 40420, 894366/5, 0, -649016238/35, ...];
n=7: [1, 48, 2205/2, 16060, 1320759/8, 6216189/5, 510096457/80, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2 * Integral 1/A(x)^2 dx)/A(x), for n > 0.
RELATED SERIES.
1/A(x) = 1 - 2*x + x^2 - 24*x^3 - 328*x^4 - 8468*x^5 - 264972*x^6 - 10068372*x^7 - 447223340*x^8 - 22709482068*x^9 - 1296038603112*x^10 + ...
exp( Integral 1/A(x)^2 dx) = 1 + x - x^2/2! - 3*x^3/3! - 135*x^4/4! - 8571*x^5/5! - 1061361*x^6/6! - 197712639*x^7/7! - 52240421007*x^8/8! - 18481482225495*x^9/9! + ...
A'(x)/A(x) = 1 + x + 37*x^2 + 753*x^3 + 22991*x^4 + 849829*x^5 + 37219617*x^6 + 1873928193*x^7 + 106404715099*x^8 + 6716223979161*x^9 + ...

Links

Crossrefs

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

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^2*intformal(1/Ser(A)^2)) / 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 = 0.0311339300124... - Vaclav Kotesovec, Oct 19 2020

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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