A305143 -id:A305143 - cp's OEIS frontend

A305137 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x) dx ) / A(x), for n > 0.

1, 1, 7, 97, 1987, 53281, 1754245, 68228209, 3055471369, 154724090845, 8740256396563, 545005932104377, 37196779826275411, 2759229671824346893, 221140447146112986889, 19051164839221523341825, 1756309610450933072328241, 172576908229287147075691417, 18010455349270266144268806799, 1989930676607696867000687913025

Offset: 0

Paul D. Hanna, May 31 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

Note: 0 = [x^n] exp( n * Integral C(x) dx ) / C(x) holds for n > 0 when C(x) = 1 + x*C(x)^2 is a g.f. of the Catalan numbers (A000108).

			O.g.f.: A(x) = 1 + x + 7*x^2 + 97*x^3 + 1987*x^4 + 53281*x^5 + 1754245*x^6 + 68228209*x^7 + 3055471369*x^8 + 154724090845*x^9 + 8740256396563*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^2*Integral A(x) dx)/A(x) begins:
n=0: [1, -1, -6, -84, -1764, -48360, -1620186, -63857556, ...];
n=1: [1, 0, -6, -88, -1830, -249144/5, -1661842, -2284994352/35, ...];
n=2: [1, 3, 0, -90, -2025, -272391/5, -8967968/5, -488439972/7, ...];
n=3: [1, 8, 30, 0, -2100, -311856/5, -10175418/5, -545952984/7, ...];
n=4: [1, 15, 114, 532, 0, -327564/5, -2389194, -3186733572/35, ...];
n=5: [1, 24, 294, 2416, 13536, 0, -2539746, -763395912/7, ...];
n=6: [1, 35, 624, 7542, 68415, 2234001/5, 0, -4102900932/35, ...];
n=7: [1, 48, 1170, 19320, 242550, 12134424/5, 90334582/5, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2*Integral A(x) dx)/A(x), for n > 0.
RELATED SERIES.
exp(Integral A(x) dx) = 1 + x + 2*x^2/2! + 18*x^3/3! + 648*x^4/4! + 50904*x^5/5! + 6700464*x^6/6! + 1310200848*x^7/7! + 354395417472*x^8/8! + 126396068810112*x^9/9! + ...
A'(x)/A(x) = 1 + 13*x + 271*x^2 + 7489*x^3 + 253771*x^4 + 10113877*x^5 + 461995381*x^6 + 23766009457*x^7 + 1359214691545*x^8 + 85572483605593*x^9 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A305138, A305139, A305140, A305141, A305142, A305143.

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

a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 0.2658290856... - Vaclav Kotesovec, Oct 19 2020

A305138 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x) dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 29, 2829, 574365, 198036917, 103683001241, 76732157303677, 76332092063105089, 98308205372662506789, 159202381329276508297845, 316697346580111339298407645, 759241108161224318180212593541, 2158996472269125350886176802590109, 7185071088438157708730937024732618961, 27665076438562174334456341670646413685101

Offset: 0

Paul D. Hanna, May 31 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

Note: 0 = [x^n] exp( n * Integral C(x) dx ) / C(x) holds for n > 0 when C(x) = 1 + x*C(x)^2 is a g.f. of the Catalan numbers (A000108).

			O.g.f.: A(x) = 1 + x + 29*x^2 + 2829*x^3 + 574365*x^4 + 198036917*x^5 + 103683001241*x^6 + 76732157303677*x^7 + 76332092063105089*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^3*Integral A(x) dx)/A(x) begins:
n=0: [1, -1, -28, -2772, -567952, -196735000, -103247834508, ...];
n=1: [1, 0, -28, -8372/3, -570045, -2957867108/15, -930709619938/9, ...];
n=2: [1, 7, 0, -8512/3, -1754242/3, -1002113658/5, -4705750542584/45, ...];
n=3: [1, 26, 323, 0, -603988, -1046990252/5, -539032922442/5, ...];
n=4: [1, 63, 1988, 119140/3, 0, -3286250072/15, -5146650533948/45, ...];
n=5: [1, 124, 7722, 961184/3, 28298144/3, 0, -1087966178572/9, ...];
n=6: [1, 215, 23192, 1672776, 90300350, 18408322114/5, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3*Integral A(x) dx)/A(x), for n > 0.
RELATED SERIES.
exp( Integral A(x) dx) = 1 + x + 2*x^2/2! + 62*x^3/3! + 17216*x^4/4! + 13870816*x^5/5! + 23847686176*x^6/6! + 74818727953312*x^7/7! + 387328675940041472*x^8/8! + ...
A'(x)/A(x) = 1 + 57*x + 8401*x^2 + 2284577*x^3 + 986920761*x^4 + 620790291801*x^5 + 536349430717661*x^6 + 610013974179245217*x^7 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A305137, A305139, A305140, A305141, A305142, A305143.

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

a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 27 / (-LambertW(-3*exp(-3)) * (3 + LambertW(-3*exp(-3)))^2) and c = 0.0614343742940117549375894089... - Vaclav Kotesovec, Oct 20 2020

A305139 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^2 dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 9, 155, 3805, 118632, 4429279, 191275884, 9340355265, 507681357635, 30360217294454, 1979895257720082, 139811654124752231, 10629630950986800850, 865954337592580081080, 75286721276048241037848, 6961094538227014053702537, 682423909436661488354778945, 70743106543492192940195723155, 7736186700358670253328879658965

Offset: 0

Paul D. Hanna, May 31 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

Note: 0 = [x^n] exp( n * Integral F(x)^2 dx ) / F(x) holds for n > 0 when F(x) = 1 + x*F(x)^3 is a g.f. of A001764.

			O.g.f.: A(x) = 1 + x + 9*x^2 + 155*x^3 + 3805*x^4 + 118632*x^5 + 4429279*x^6 + 191275884*x^7 + 9340355265*x^8 + 507681357635*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^2*Integral A(x)^2 dx)/A(x) begins:
n=0: [1, -1, -8, -138, -3440, -108905, -4118952, -179740162, ...];
n=1: [1, 0, -15/2, -140, -28065/8, -111009, -67094895/16, -1279321635/7, ...];
n=2: [1, 3, 0, -130, -3660, -117117, -4419200, -1344147030/7, ...];
n=3: [1, 8, 65/2, 0, -27265/8, -124886, -76650687/16, -2911952885/14, ...];
n=4: [1, 15, 120, 630, 0, -117081, -5202600, -1614205230/7, ...];
n=5: [1, 24, 609/2, 2712, 139455/8, 0, -78693087/16, -3562210803/14, ...];
n=6: [1, 35, 640, 8190, 81940, 620323, 0, -1698895510/7, ...];
n=7: [1, 48, 2385/2, 20540, 2215455/8, 3088737, 428675377/16, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2*Integral A(x)^2 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 19*x^2 + 328*x^3 + 8001*x^4 + 247664*x^5 + 9188337*x^6 + 394725252*x^7 + 19194243265*x^8 + 1039762257722*x^9 + ...
exp( Integral A(x)^2 dx) = 1 + x + 3*x^2/2! + 45*x^3/3! + 2145*x^4/4! + 203085*x^5/5! + 30980475*x^6/6! + 6838973145*x^7/7! + 2045481775425*x^8/8! + 792696897387225*x^9/9! + ...
A'(x)/A(x) = 1 + 17*x + 439*x^2 + 14473*x^3 + 568296*x^4 + 25625759*x^5 + 1297831032*x^6 + 72732570537*x^7 + 4462331350255*x^8 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A305137, A305138, A305140, A305141, A305142, A305143.

{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(Ser(A)^2)) / Ser(A) )[m+1] );A[n+1]}
for(n=0,20,print1(a(n),", "))

a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 0.652748826558... - Vaclav Kotesovec, Oct 19 2020

A305140 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x)^2 dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 33, 3422, 710395, 245288190, 127281447538, 92967363233586, 91202509214139831, 115939599286159123295, 185623891076803234259504, 365706330842590993758556662, 869715472542563657980211015186, 2456766458611829222907737567821138, 8131203421875726862447708824758000364

Offset: 0

Paul D. Hanna, May 31 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

Note: 0 = [x^n] exp( n * Integral F(x)^2 dx ) / F(x) holds for n > 0 when F(x) = 1 + x*F(x)^3 is a g.f. of A001764.

			O.g.f.: A(x) = 1 + x + 33*x^2 + 3422*x^3 + 710395*x^4 + 245288190*x^5 + 127281447538*x^6 + 92967363233586*x^7 + 91202509214139831*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^3*Integral A(x)^2 dx)/A(x) begins:
n=0: [1, -1, -32, -3357, -702560, -243654950, ...];
n=1: [1, 0, -63/2, -3367, -5633901/8, -2440775421/10, ...];
n=2: [1, 7, 0, -3325, -715316, -1235155194/5, ...];
n=3: [1, 26, 665/2, 0, -5720533/8, -5095053859/20, ...];
n=4: [1, 63, 2016, 41699, 0, -1290302622/5, ...];
n=5: [1, 124, 15561/2, 328643, 80013395/8, 0, ...];
n=6: [1, 215, 23296, 1697283, 93264388, 19574613422/5, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3*Integral A(x)^2 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 67*x^2 + 6910*x^3 + 1428723*x^4 + 492223022*x^5 + 255112067610*x^6 + 186210340326168*x^7 + 182601537143712727*x^8 + ...
exp( Integral A(x)^2 dx) = 1 + x + 5*x^2/2! + 415*x^3/3! + 167521*x^4/4! + 172296341*x^5/5! + 355443416701*x^6/6! + 1288266047868955*x^7/7! + 7518341623369166465*x^8/8! + ...
A'(x)/A(x) = 1 + 65*x + 10168*x^2 + 2825845*x^3 + 1222346736*x^4 + 762046826846*x^5 + 649809039848130*x^6 + 728835192043655757*x^7 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A305137, A305138, A305139, A305141, A305142, A305143.

{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(Ser(A)^2)) / Ser(A) )[m+1] );A[n+1]}
for(n=0,20,print1(a(n),", "))

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

A305142 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x)^3 dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 37, 4096, 878619, 306873869, 158938884952, 114993958088544, 111352808890827351, 139608635486408132803, 220605354590414591998297, 429593550416513276960527556, 1011544195064396609819653321932, 2833764097327349890282080026444076, 9314700709596523207841989131758528948, 35498787449426898120781594428097022541008

Offset: 0

Paul D. Hanna, May 31 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

Note: 0 = [x^n] exp( n * Integral F(x)^3 dx ) / F(x) holds for n > 0 when F(x) = 1 + x*F(x)^4 is a g.f. of A002293.

			O.g.f.: A(x) = 1 + x + 37*x^2 + 4096*x^3 + 878619*x^4 + 306873869*x^5 + 158938884952*x^6 + 114993958088544*x^7 + 111352808890827351*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^3*Integral A(x)^3 dx)/A(x) begins:
  n=0: [1, -1, -36, -4023, -869168, -304829775, ...];
  n=1: [1, 0, -35, -12064/3, -870135, -915526348/3, ...];
  n=2: [1, 7, 0, -11609/3, -2626022/3, -307526817, ...];
  n=3: [1, 26, 342, 0, -847892, -312911550, ...];
  n=4: [1, 63, 2044, 131387/3, 0, -919948381/3, ...];
  n=5: [1, 124, 7839, 1011556/3, 31877746/3, 0, ...];
  n=6: [1, 215, 23400, 1722357, 96411130, 4177156347, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3*Integral A(x)^3 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 75*x^2 + 8266*x^3 + 1766799*x^4 + 615808080*x^5 + 318573312664*x^6 + 230335700260146*x^7 + 222950653057400247*x^8 + ...
A(x)^3 = 1 + 3*x + 114*x^2 + 12511*x^3 + 2664651*x^4 + 926819028*x^5 + 478906878958*x^6 + 346026409343751*x^7 + 334794104506072215*x^8 + ...
exp( Integral A(x)^3 dx) = 1 + x + 7*x^2/2! + 703*x^3/3! + 303145*x^4/4! + 321307921*x^5/5! + 669264720031*x^6/6! + 2418416266536607*x^7/7! + 13971240948079459633*x^8/8! + ...
A'(x)/A(x) = 1 + 73*x + 12178*x^2 + 3495501*x^3 + 1529245631*x^4 + 951553836400*x^5 + 803743212623394*x^6 + 889843851811684197*x^7 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A305137, A305138, A305139, A305140, A305141, A305143.

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

a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 27 / (-LambertW(-3*exp(-3)) * (3 + LambertW(-3*exp(-3)))^2) and c = 0.0710327332647009858916047504... - Vaclav Kotesovec, Oct 20 2020

A305141 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^3 dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 11, 228, 6621, 240689, 10351550, 509604000, 28110904439, 1711981045939, 113863658640249, 8201890764752000, 635637023178406472, 52712939749766528868, 4656568244615480818794, 436486181882215383918344, 43268184144892865821692559, 4522468113281674174052795751, 497107356171097228291772997005

Offset: 0

Paul D. Hanna, May 31 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

Note: 0 = [x^n] exp( n * Integral F(x)^3 dx ) / F(x) holds for n > 0 when F(x) = 1 + x*F(x)^4 is a g.f. of A002293.

			O.g.f.: A(x) = 1 + x + 11*x^2 + 228*x^3 + 6621*x^4 + 240689*x^5 + 10351550*x^6 + 509604000*x^7 + 28110904439*x^8 + 1711981045939*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^2*Integral A(x)^3 dx)/A(x) begins:
n=0: [1, -1, -10, -207, -6076, -223435, -9707184, ...];
n=1: [1, 0, -9, -616/3, -6115, -1128624/5, -88359418/9, ...];
n=2: [1, 3, 0, -535/3, -6107, -1156806/5, -455986832/45, ...];
n=3: [1, 8, 35, 0, -5257, -1167296/5, -52842348/5, ...];
n=4: [1, 15, 126, 2219/3, 0, -1003419/5, -96971176/9, ...];
n=5: [1, 24, 315, 9104/3, 22299, 0, -83502496/9, ...];
n=6: [1, 35, 656, 8883, 98045, 4304146/5, 0, ...];
n=7: [1, 48, 1215, 65480/3, 316393, 19736784/5, 1805083618/45, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2*Integral A(x)^3 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 23*x^2 + 478*x^3 + 13819*x^4 + 499636*x^5 + 21382124*x^6 + 1048225434*x^7 + 57622342803*x^8 + 3499302699294*x^9 + ...
A(x)^3 = 1 + 3*x + 36*x^2 + 751*x^3 + 21627*x^4 + 777888*x^5 + 33127964*x^6 + 1617262071*x^7 + 88594431639*x^8 + 5364836605107*x^9 + ...
exp( Integral A(x)^3 dx) = 1 + x + 7*x^2/2! + 235*x^3/3! + 19033*x^4/4! + 2701081*x^5/5! + 578096911*x^6/6! + 171419630467*x^7/7! + 66700397369425*x^8/8! + ...
A'(x)/A(x) = 1 + 21*x + 652*x^2 + 25373*x^3 + 1159491*x^4 + 60142320*x^5 + 3468823324*x^6 + 219440572309*x^7 + 15077173544671*x^8 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A305137, A305138, A305139, A305140, A305142, A305143.

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

a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 2.1981... - Vaclav Kotesovec, Oct 19 2020

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

Paul D. Hanna, May 31 2018

nonn

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.

			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 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..520

Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.

Cf. A305145, A305146, A305147, A304861, A304862.

{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), ", "))

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

Paul D. Hanna, May 31 2018

nonn

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.

			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 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..198

Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.

Cf. A305144, A305146, A305147, A304861, A304862.

{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), ", "))

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

Paul D. Hanna, May 31 2018

nonn

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.

			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 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..144

Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.

Cf. A305144, A305145, A305147.

{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), ", "))

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

Paul D. Hanna, May 31 2018

nonn

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.

			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 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..344

Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.

Cf. A305144, A305145, A305146.

{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), ", "))

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

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A305137 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x) dx ) / A(x), for n > 0.

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A305138 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x) dx ) / A(x), for n > 0.

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A305139 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^2 dx ) / A(x), for n > 0.

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A305140 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x)^2 dx ) / A(x), for n > 0.

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A305142 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x)^3 dx ) / A(x), for n > 0.

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A305141 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^3 dx ) / A(x), for n > 0.

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

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

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