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 1-7 of 7 results.

A337458 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1) * x ) / A(x)^(n+1) = 0 for n > 0.

Original entry on oeis.org

1, 1, 2, 11, 130, 2450, 63012, 2040779, 79377914, 3594766694, 185457776252, 10725423627006, 686721189003668, 48200778475446916, 3679104677398632520, 303348177377608050219, 26865664102518601306154, 2543352040870175109554654, 256296085507636954980717708, 27390678829206902911266889386
Offset: 0

Views

Author

Paul D. Hanna, Aug 28 2020

Keywords

Comments

It is remarkable that this sequence consists entirely of integers.

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 130*x^4 + 2450*x^5 + 63012*x^6 + 2040779*x^7 + 79377914*x^8 + 3594766694*x^9 + 185457776252*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+1)*x) / A(x)^(n+1) begins:
n=0: [1, -1, -2, -48, -2616, -262080, -41718240, -9630270720, ...];
n=1: [1, 0, -6, -112, -5592, -547968, -86345120, -19809990912, ...];
n=2: [1, 3, 0, -222, -10728, -958824, -144971712, -32519314080, ...];
n=3: [1, 8, 52, 0, -18648, -1693248, -236690784, -50727983616, ...];
n=4: [1, 15, 210, 2420, 0, -2739720, -399251600, -80125144800, ...];
n=5: [1, 24, 558, 12192, 221184, 0, -616918320, -131299591680, ...];
n=6: [1, 35, 1204, 40278, 1272768, 33597312, 0, -196436730672, ...];
n=7: [1, 48, 2280, 106688, 4869552, 210771456, 7654459648, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+1)*x ) / A(x)^(n+1) = 0 for n>0.
RELATED SERIES.
Define B(x) = A(x/B(x)), which begins
B(x) = 1 + x + x^2 + 7*x^3 + 93*x^4 + 1859*x^5 + 49357*x^6 + 1629227*x^7 + 64149805*x^8 + 2929386667*x^9 + ... + A337457(n)*x^n + ...
then the table of coefficients of x^k/k! in exp( n*(n-1)*x/B(x) ) begins:
n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
n=2: [1, 2, 0, -16, -320, -21888, -2648576, -494325760, ...];
n=3: [1, 6, 24, 0, -1728, -88704, -9621504, -1715198976, ...];
n=4: [1, 12, 120, 864, 0, -281088, -26873856, -4328017920, ...];
n=5: [1, 20, 360, 5600, 65920, 0, -66944000, -10207436800, ...];
n=6: [1, 30, 840, 21600, 492480, 8784000, 0, -22098355200, ...];
n=7: [1, 42, 1680, 63504, 2237760, 71229312, 1814690304, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n-1)*x/B(x) ) = 0 for n>0.
Also note that B(x) = x/Series_Reversion( x*A(x) ) and A(x) = B(x*A(x)).

Links

Crossrefs

Cf. A337457, A304319.

Programs

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

Formula

Given o.g.f. A(x), define B(x) = A(x/B(x)), then B(x) is the o.g.f. of A337457 and satisfies [x^n] exp( n*(n-1)*x/B(x) ) = 0 for n>0.
a(n) ~ sqrt(1-c) * 2^(2*n - 3/2) * n^(n - 3/2) / (sqrt(Pi) * c^n * (2-c)^(n-1) * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

A304318 O.g.f. A(x) satisfies: [x^n] exp( n*(n-1) * x ) / A(x) = 0.

Original entry on oeis.org

1, 0, 2, 24, 436, 10656, 328112, 12183456, 529242224, 26309617536, 1472135847072, 91526938123008, 6258004268952064, 466599240364076544, 37672137946943244288, 3274012281487011586560, 304724394621209905647360, 30239686358027369113804800, 3187164738879981461171955200, 355548230503664593634743375872
Offset: 0

Views

Author

Paul D. Hanna, May 11 2018

Keywords

Comments

It is striking that the coefficients of o.g.f. A(x) consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + 2*x^2 + 24*x^3 + 436*x^4 + 10656*x^5 + 328112*x^6 + 12183456*x^7 + 529242224*x^8 + 26309617536*x^9 + 1472135847072*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n-1)*x) / A(x) begins:
n=0: [1, 0, -4, -144, -10368, -1267200, -234576000, -61085767680, ...];
n=1: [1, 0, -4, -144, -10368, -1267200, -234576000, -61085767680, ...];
n=2: [1, 2, 0, -160, -11600, -1376928, -250428416, -64479262720, ...];
n=3: [1, 6, 32, 0, -13392, -1630944, -286447104, -71981250048, ...];
n=4: [1, 12, 140, 1440, 0, -1916928, -351444096, -85338800640, ...];
n=5: [1, 20, 396, 7616, 128512, 0, -417488000, -107269127680, ...];
n=6: [1, 30, 896, 26496, 760752, 19101600, 0, -128348167680, ...];
n=7: [1, 42, 1760, 73440, 3034800, 121743072, 4260708864, 0, ...];
n=8: [1, 56, 3132, 174800, 9716608, 535021056, 28597069696, 1331047703552, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n-1)*x ) / A(x) = 0 for n>=0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304316:
A'(x)/A(x) = 4*x + 72*x^2 + 1736*x^3 + 53040*x^4 + 1961728*x^5 + 85062432*x^6 + 4225904800*x^7 + 236455369344*x^8 + 14705880874944*x^9 + 1005982098054912*x^10 + ... + A304316(n)*x^n + ...

Links

Crossrefs

Cf. A304316, A304319, A304320.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m-2) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0,25, print1( a(n),", "))

Formula

a(n) ~ sqrt(1-c) * 2^(2*n - 1) * n^(n - 1/2) / (sqrt(Pi) * c^(n - 1/2) * (2-c)^n * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

A304317 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n*(n+1) * x ) / F(x) = 0 for n>0.

Original entry on oeis.org

2, 16, 260, 6200, 191832, 7235152, 320372320, 16243028896, 926219213216, 58608051937536, 4072302306624576, 308103163039452416, 25206121081157341184, 2216835666085110716928, 208552041718766935142400, 20896682922005650384144896, 2221700463749100463405564416, 249800738062720558095843241984, 29615243677328447562465854639104
Offset: 0

Views

Author

Paul D. Hanna, May 11 2018

Keywords

Examples

			O.g.f.: L(x) = 2 + 16*x + 260*x^2 + 6200*x^3 + 191832*x^4 + 7235152*x^5 + 320372320*x^6 + 16243028896*x^7 + 926219213216*x^8 + 58608051937536*x^9 + 4072302306624576*x^10 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304319:
F(x) = 1 + 2*x + 10*x^2 + 104*x^3 + 1772*x^4 + 42408*x^5 + 1303504*x^6 + 48736000*x^7 + 2139552016*x^8 + 107629121888*x^9 + 6094743943584*x^10 + ... + A304319(n)*x^n + ...
which satisfies [x^n] exp( n*(n+1) * x ) / F(x) = 0 for n>0.

Links

Crossrefs

Cf. A304319, A304316, A304321.

Programs

  • Mathematica
    m = 25;
F = 1 + Sum[c[k] x^k, {k, m}];
s[n_] := Solve[SeriesCoefficient[Exp[n*(n + 1)*x]/F, {x, 0, n}] == 0][[1]];
Do[F = F /. s[n], {n, m}];
CoefficientList[D[F, x]/F + O[x]^m, x] (* Jean-François Alcover, May 21 2018 *)
  • PARI
    {a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m*(m-1) +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]}
for(n=0,25, print1( a(n),", "))

Formula

Logarithmic derivative of the o.g.f. of A304319.
a(n) ~ sqrt(1-c) * 2^(2*n + 2) * n^(n + 3/2) / (sqrt(Pi) * c^(n + 3/2) * (2-c)^(n+1) * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

A304863 O.g.f. A(x) satisfies: [x^n] exp( n*(n+2) * x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 3, 17, 180, 3079, 74271, 2308940, 87438684, 3888682559, 198073751505, 11348409001233, 721483807171188, 50361931297722244, 3827114191186713588, 314413091556481490640, 27761835704580647457012, 2621495363274661266785679, 263593068966612639018287637, 28117066903131481643928647363, 3171150259810035292799245555884, 377044852592342586608552585592079
Offset: 0

Views

Author

Paul D. Hanna, May 19 2018

Keywords

Examples

			O.g.f.: A(x) = 1 + 3*x + 17*x^2 + 180*x^3 + 3079*x^4 + 74271*x^5 + 2308940*x^6 + 87438684*x^7 + 3888682559*x^8 + 198073751505*x^9 + 11348409001233*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+2)*x) / A(x) begins:
n=0: [1, -3, -16, -630, -50112, -6665400, -1325134080, -366674636160, ...];
n=1: [1, 0, -25, -828, -58779, -7479072, -1452239685, -395811449100, ...];
n=2: [1, 5, 0, -1078, -78464, -9183672, -1700942720, -450843184000, ...];
n=3: [1, 12, 119, 0, -99387, -12381300, -2151101205, -544666984560, ...];
n=4: [1, 21, 416, 6858, 0, -15533496, -2923952256, -703585823616, ...];
n=5: [1, 32, 999, 29540, 730213, 0, -3653179205, -962999633260, ...];
n=6: [1, 45, 2000, 86922, 3589056, 124275528, 0, -1200826684800, ...];
n=7: [1, 60, 3575, 210672, 12162501, 668679228, 30900268395, 0, ...];
n=8: [1, 77, 5904, 449930, 33949888, 2513449800, 177544721920, 10559736679040, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+2)*x ) / A(x) = 0 for n>=0.
RELATED SERIES.
The logarithmic derivative of A(x) yields:
A'(x)/A(x) = 3 + 25*x + 414*x^2 + 10109*x^3 + 320253*x^4 + 12346720*x^5 + 557708406*x^6 + 28786882117*x^7 + 1668054884229*x^8 + 107077380781005*x^9 + ...
1 - 1/A(x) = 3*x + 8*x^2 + 105*x^3 + 2088*x^4 + 55545*x^5 + 1840464*x^6 + 72752904*x^7 + 3334122880*x^8 + 173569203225*x^9 + 10108800765000*x^10 + ...

Crossrefs

Cf. A304322, A304318, A304319, A304864, A304865.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m+1) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

A304864 O.g.f. A(x) satisfies: [x^n] exp( n*(n+3) * x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 4, 26, 288, 5012, 122608, 3869456, 148838816, 6721823600, 347434618432, 20180665251360, 1299399587904384, 91769604540962816, 7049102617933604352, 584848346900868001792, 52109481481410100183552, 4961586770799906448318208, 502707358017324652042259456, 54000226687663791374322245120, 6129804668943947684749062516736, 733179029209444818691965317379072
Offset: 0

Views

Author

Paul D. Hanna, May 19 2018

Keywords

Examples

			O.g.f.: A(x) = 1 + 4*x + 26*x^2 + 288*x^3 + 5012*x^4 + 122608*x^5 + 3869456*x^6 + 148838816*x^7 + 6721823600*x^8 + 347434618432*x^9 + 20180665251360*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+3)*x) / A(x) begins:
n=0: [1, -4, -20, -864, -72576, -10060800, -2068882560, -589017945600, ...];
n=1: [1, 0, -36, -1232, -89088, -11667456, -2328963200, -650497926144, ...];
n=2: [1, 6, 0, -1664, -125136, -14853600, -2803074560, -757869964800, ...];
n=3: [1, 14, 160, 0, -162000, -20768352, -3651775488, -937273259520, ...];
n=4: [1, 24, 540, 10000, 0, -26468352, -5107476608, -1241737082880, ...];
n=5: [1, 36, 1260, 41536, 1133184, 0, -6460818560, -1740188582400, ...];
n=6: [1, 50, 2464, 118368, 5374512, 202760544, 0, -2192436486144, ...];
n=7: [1, 66, 4320, 279136, 17619504, 1054101600, 52553405440, 0, ...];
n=8: [1, 84, 7020, 582400, 47760000, 3832731648, 292170316672, 18603667330560, 0, ...]; ...
RELATED SERIES.
The logarithmic derivative of A(x) yields:
A'(x)/A(x) = 4 + 36*x + 616*x^2 + 15496*x^3 + 504624*x^4 + 19947072*x^5 + 921521248*x^6 + 48536700064*x^7 + 2864002270720*x^8 + 186878075521216*x^9 + ...
1 - 1/A(x) = 4*x + 10*x^2 + 144*x^3 + 3024*x^4 + 83840*x^5 + 2873448*x^6 + 116868640*x^7 + 5488631808*x^8 + 291890096640*x^9 + 17321970359200*x^10 + ...

Crossrefs

Cf. A304322, A304318, A304319, A304863, A304865.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m+2) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

A304865 O.g.f. A(x) satisfies: [x^n] exp( n*(n+4) * x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 5, 37, 434, 7751, 193101, 6200872, 242667316, 11144759839, 585318934391, 34511211188637, 2253285826341378, 161201686356627524, 12530568505972885004, 1051099249634285619168, 94603882448795669308980, 9092091650779263675187695, 929177036869575506758681035, 100608724821944458615599713935, 11504982932704269804549116593702, 1385525417578463389730054278506959
Offset: 0

Views

Author

Paul D. Hanna, May 19 2018

Keywords

Examples

			O.g.f.: A(x) = 1 + 5*x + 37*x^2 + 434*x^3 + 7751*x^4 + 193101*x^5 + 6200872*x^6 + 242667316*x^7 + 11144759839*x^8 + 585318934391*x^9 + 34511211188637*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+4)*x) / A(x) begins:
n=0: [1, -5, -24, -1134, -100608, -14542200, -3095496000, -907608905280, ...];
n=1: [1, 0, -49, -1744, -128763, -17383400, -3572628125, -1024052930280, ...];
n=2: [1, 7, 0, -2430, -189600, -22895928, -4410982656, -1218708054720, ...];
n=3: [1, 16, 207, 0, -250107, -33107544, -5910144669, -1540910769048, ...];
n=4: [1, 27, 680, 13970, 0, -42775928, -8486494016, -2090851421760, ...];
n=5: [1, 40, 1551, 56376, 1681797, 0, -10852876125, -2994692165280, ...];
n=6: [1, 55, 2976, 156546, 7748832, 316211400, 0, -3807596825280, ...];
n=7: [1, 72, 5135, 360920, 24718725, 1597879072, 85448027299, 0, ...];
n=8: [1, 91, 8232, 738450, 65376768, 5650680456, 462123838656, 31350065660352, 0, ...]; ...
RELATED SERIES.
The logarithmic derivative of A(x) yields:
A'(x)/A(x) = 5 + 49*x + 872*x^2 + 22661*x^3 + 759915*x^4 + 30843448*x^5 + 1459277062*x^6 + 78529473925*x^7 + 4724556111179*x^8 + 313739794874469*x^9 + ...
1 - 1/A(x) = 5*x + 12*x^2 + 189*x^3 + 4192*x^4 + 121185*x^5 + 4299300*x^6 + 180081132*x^7 + 8675950464*x^8 + 471853727865*x^9 + 28563862383700*x^10 + ...

Crossrefs

Cf. A304322, A304318, A304319, A304863, A304864.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m+3) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

A383559 O.g.f. A(x) satisfies: [x^n] exp( n*(2*n+1)*x ) / A(x) = 0 for n > 0.

Original entry on oeis.org

1, 3, 29, 609, 20857, 997671, 61114409, 4548317073, 397323349505, 39774233809179, 4483232458612245, 561425116837715457, 77289022946177141161, 11597365849594347661839, 1883429636306366952452433, 329083700898584984268782241, 61549497773760817234065857793, 12268604214374346472111552473267
Offset: 0

Views

Author

Paul D. Hanna, May 17 2025

Keywords

Examples

			O.g.f.: A(x) = 1 + 3*x + 29*x^2 + 609*x^3 + 20857*x^4 + 997671*x^5 + 61114409*x^6 + 4548317073*x^7 + 397323349505*x^8 + ...
RELATED TABLE.
The table of coefficients of x^k/k! in exp( n*(2*n+1)*x ) / A(x) begins
  n = 1: [1,  0,  -49,  -3186,  -445203, -109403892, -41045026725, ...];
  n = 2: [1,  7,    0,  -3872,  -546416, -126698400, -45990717440, ...];
  n = 3: [1, 18,  275,      0,  -664875, -160762914, -55439093817, ...];
  n = 4: [1, 33, 1040,  27900,        0, -196031664, -71849477952, ...];
  n = 5: [1, 52, 2655, 129778,  5408749,          0, -87799444565, ...];
  n = 6: [1, 75, 5576, 407664, 28585872, 1710760608,            0, ...];
  ...
illustrating [x^n] exp( n*(2*n+1)*x ) / A(x) = 0 for n > 0.

Links

Crossrefs

Cf. A304319, A337458.

Programs

  • PARI
    {a(n) = my(A=[1],m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(2*m-1) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

Formula

a(n) ~ sqrt(1-w) * 2^(3*n - 1/4) * n^(n - 1/2) / (sqrt(Pi) * exp(n) * (2-w)^n * w^(n + 1/4)), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, May 18 2025
Showing 1-7 of 7 results.

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