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 1-6 of 6 results.

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

A305146 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^4 * Integral 1/A(x) dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 105, 71030, 143839875, 639147831054, 5268190256643730, 72401453092661090460, 1539974714406342828684915, 47967103851505667222316762710, 2096230585920937730055252273554166, 124208697361885403106994025647669349700, 9703933918967416448770462097241544278503550, 976615535896268261227542752682139965289070564940
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 + 105*x^2 + 71030*x^3 + 143839875*x^4 + 639147831054*x^5 + 5268190256643730*x^6 + 72401453092661090460*x^7 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^4*Integral 1/A(x) dx)/A(x) begins:
n=0: [1, -1, -104, -70821, -143687104, -638845480750, -5266877186423376, ...];
n=1: [1, 0, -105, -70960, -143775630, -639017901600, -5267622501808905, ...];
n=2: [1, 15, 0, -72605, -145123140, -641617076562, -5278826440840960, ...];
n=3: [1, 80, 3055, 0, -149843050, -653149632064, -5327910150826725, ...];
n=4: [1, 255, 32280, 2624475, 0, -678395417454, -5464268996914000, ...];
n=5: [1, 624, 194271, 40142304, 6023531646, 0, -5698446198253501, ...];
n=6: [1, 1295, 837760, 360867555, 116236431740, 29089429020014, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^4 * Integral 1/A(x) dx)/A(x), for n > 0.
RELATED SERIES.
1/A(x) = 1 - x - 104*x^2 - 70821*x^3 - 143687104*x^4 - 638845480750*x^5 - 5266877186423376*x^6 - 72390764082089330493*x^7 + ...
exp( Integral 1/A(x) dx) = 1 + x - 35*x^3 - 17740*x^4 - 28755126*x^5 - 106502983600*x^6 - 752517500258415*x^7 - 9049597920124635300*x^8 - 171101127726280225469450*x^9 + ..., which is an integer series.
A'(x)/A(x) = 1 + 209*x + 212776*x^2 + 575053749*x^3 + 3194983074896*x^4 + 31605201852299630*x^5 + 506772757749658101024*x^6 + 12319213675791316095636957*x^7 + ...

Links

Crossrefs

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

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^4*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^(8*n - 5/2) * n^(3*n - 1/2) / (sqrt(Pi) * exp(3*n) * c^n * (4-c)^(3*n - 1)), where c = -LambertW(-4*exp(-4)) = 0.079309605127113656439108647... - 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

A229044 G.f. A(x) satisfies: [x^(n+1)] A(x)^(n^2) = 0 for n>=0.

Original entry on oeis.org

1, 1, 0, -1, -6, -78, -1544, -40605, -1328178, -51857806, -2350025232, -121120896906, -6991877399100, -446673990116508, -31277285155060464, -2381645560450404989, -195914136385421694954, -17312472044077536945630, -1635541992950202705979424, -164494265246550280147797438
Offset: 0

Views

Author

Paul D. Hanna, Sep 12 2013

Keywords

Examples

			G.f.: A(x) = 1 + x - x^3 - 6*x^4 - 78*x^5 - 1544*x^6 - 40605*x^7 -...
Coefficients of x^k in the square powers A(x)^(n^2) of g.f. A(x) begin:
n=1: [1, 1,   0,   -1,    -6,    -78,   -1544,   -40605,  -1328178, ...];
n=2: [1, 4,   6,    0,   -35,   -396,   -7182,  -181824,  -5817510, ...];
n=3: [1, 9,  36,   75,     0,  -1260,  -21408,  -499203, -15299145, ...];
n=4: [1,16, 120,  544,  1484,      0,  -52656, -1202240, -34269906, ...];
n=5: [1,25, 300, 2275, 11900,  40680,       0, -2557775, -73526475, ...];
n=6: [1,36, 630, 7104, 57429, 345204, 1430418,        0,-142432290, ...];
n=7: [1,49,1176,18375,209230,1833678,12546744, 61418175,         0, ...];
n=8: [1,64,2016,41600,630960,7470336,71271616,549420288,3113335320, 0, ...]; ...
where the coefficients of x^(n+1) in A(x)^(n^2) all equal zero for n>=0.
Related expansions.
A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) is the g.f. of A185072:
G(x) = 1 + x - 2*x^2 + 6*x^3 - 28*x^4 + 70*x^5 - 1446*x^6 -...
A(x)'/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 - ...
A(x)/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 + ... + A305144(n)*x^n + ...

Crossrefs

Cf. A185072, A305144, A230218, A229041, A171791.

Programs

  • PARI
    {a(n)=local(A=[1,1]);for(k=1,n,A=concat(A,0);A[#A]=-polcoeff((Ser(A) +O(x^(k+2)))^(k^2)/(k^2),k+1));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

a(n) is odd iff n+1 is a power of 2 (conjecture).
G.f. A(x) satisfies the following relationes.
(1) [x^(n+1)] A(x)^(n^2) = 0 for n>=0.
(2) A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) is the g.f. of A185072.
(3) A(x)/A(x)' is the g.f. of A305144. - Paul D. Hanna, Oct 23 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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