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.

Previous Showing 11-17 of 17 results.

A300617 O.g.f. A(x) satisfies: [x^n] exp( n * A(x) ) = n^2 * [x^(n-1)] exp( n * A(x) ) for n>=1.

Original entry on oeis.org

1, 3, 30, 550, 15375, 601398, 31299268, 2093655600, 175312873125, 17987972309725, 2221603804365924, 325310016974127276, 55749742122979646105, 11056914755618659399500, 2513208049272148754203200, 649086459674801585681092992, 189044817293654530855544266209, 61671809408989968268084102641075, 22399957973327602630210233608217250, 9009223131975798265447660437783058050
Offset: 1

Views

Author

Paul D. Hanna, Mar 10 2018

Keywords

Comments

Compare to: [x^n] exp( n * x ) = [x^(n-1)] exp( n * x ) for n>=1.
It is conjectured that this sequence consists entirely of integers.
a(n) is divisible by n*(n+1)/2 (conjecture); A300589(n) = a(n) / (n*(n+1)/2).

Examples

			O.g.f.: A(x) = x + 3*x^2 + 30*x^3 + 550*x^4 + 15375*x^5 + 601398*x^6 + 31299268*x^7 + 2093655600*x^8 + 175312873125*x^9 + 17987972309725*x^10 + ...
where
exp(A(x)) = 1 + x + 7*x^2/2! + 199*x^3/3! + 14065*x^4/4! + 1924201*x^5/5! + 445859911*x^6/6! + 161145717727*x^7/7! + 85790577700129*x^8/8! + ... + A300616(n)*x^n/n! + ...
such that: [x^n] exp( n * A(x) ) = n^2 * [x^(n-1)] exp( n * A(x) ).
RELATED SEQUENCES.
The sequence A300589(n) = a(n) / (n*(n+1)/2) begins:
[1, 1, 5, 55, 1025, 28638, 1117831, 58157100, 3895841625, 327054041995, ...].
The table of coefficients in x^k/k! in exp(-n*A(x)) * (1 - n^2*x) begins:
n=1: [1, 0, 5, 178, 13269, 1853876, 434314705, 158024698350, ...];
n=2: [1, -2, 0, 248, 22976, 3416592, 822150016, 303575549440, ...];
n=3: [1, -6, -27, 0, 21861, 4129758, 1079984097, 415322613324, ...];
n=4: [1, -12, -88, -848, 0, 3286304, 1109402752, 469332346368, ...];
n=5: [1, -20, -195, -2650, -55675, 0, 794678425, 438768342850, ...];
n=6: [1, -30, -360, -5832, -161856, -6828624, 0, 293555007360, ...];
n=7: [1, -42, -595, -10892, -339339, -18549958, -1433676839, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.

Links

Crossrefs

Cf. A300616, A300589, A296171, A300591, A300593, A300595, A300597.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^2*V[#A-1] - V[#A])/(#A-1) ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = -Vec( exp(m^1*x*Ser(A))*(1-m^2*x +x^2*O(x^m))^(1))[m+1]/m ); A[n]}
for(n=1, 20, print1(a(n), ", "))

Formula

O.g.f. equals the logarithm of the e.g.f. of A300616.
O.g.f. A(x) satisfies: [x^n] exp(-n*A(x)) * (1 - n^2*x) = 0, for n > 0. - Paul D. Hanna, Oct 15 2018
a(n) ~ c * (n!)^2, where c = 1.685041722777551007711429045295022018562828... - Vaclav Kotesovec, Mar 10 2018

A296177 G.f. equals the logarithm of the e.g.f. of A296176.

Original entry on oeis.org

1, -15, -6090, -30600650, -593306350650, -31192838317208826, -3652177141294409632400, -836986399841753367052602000, -342157863774785896821739864893375, -232492750600387706453977026534258393375, -248374508240426643818180115122847840121356750, -398845502818641863837604075681689663598753652620750
Offset: 1

Views

Author

Paul D. Hanna, Dec 07 2017

Keywords

Comments

E.g.f. G(x) of A296176 satisfies: [x^(n-1)] G(x)^(n^5) = [x^n] G(x)^(n^5) for n>=1.

Examples

			G.f. A(x) = x - 15*x^2 - 6090*x^3 - 30600650*x^4 - 593306350650*x^5 - 31192838317208826*x^6 - 3652177141294409632400*x^7 - 836986399841753367052602000*x^8 - 342157863774785896821739864893375*x^9 - 232492750600387706453977026534258393375*x^10 +...
such that
G(x) = exp(A(x)) = 1 + x - 29*x^2/2! - 36629*x^3/3! - 734559239*x^4/4! - 71200423546199*x^5/5! - 22459270436075644469*x^6/6! - 18407129959728493123679069*x^7/7! - 33747438879000326056232288023439*x^8/8! - 124162549312926509293620790889452447919*x^9/9! - 843670934957017748849439817665935283173590349*x^10/10! +...
satisfies [x^(n-1)] G(x)^(n^5) = [x^n] G(x)^(n^5) for n>=1.
Series_Reversion(A(x)) = x + 15*x^2 + 6540*x^3 + 31074275*x^4 + 596201157450*x^5 + 31256650109242326*x^6 + 3655957957134009767520*x^7 + 837481638576442353884460435*x^8 +...

Links

Crossrefs

Cf. A296176, A296171, A296173, A296175.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^5)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^5 ); polcoeff(log(Ser(A)),n)}
for(n=1,30,print1(a(n),", "))

A300871 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1) * A(x) ) = n*(n+1) * [x^(n-1)] exp( n*(n+1) * A(x) ) for n>=1.

Original entry on oeis.org

1, 3, 48, 1510, 71280, 4511808, 361640832, 35516910960, 4184770003200, 582762638275840, 94800017774905344, 17836975939663156224, 3847898790157443653632, 944223655310222217584640, 261663903298936561335828480, 81353978185283974468642093056, 28208743160867030634605718994944, 10849126423364041648181194666082304, 4605289001051501407092469612444385280
Offset: 1

Views

Author

Paul D. Hanna, Mar 14 2018

Keywords

Comments

Compare to: [x^n] exp( n*(n+1) * x ) = (n+1) * [x^(n-1)] exp( n*(n+1) * x ) for n>=1.
O.g.f. equals the logarithm of the e.g.f. of A300870.
The e.g.f. G(x) of A300870 satisfies: [x^n] G(x)^(n*(n+1)) = n*(n+1) * [x^(n-1)] G(x)^(n*(n+1)) for n>=1.
It is conjectured that this sequence consists entirely of integers.
a(n) is divisible by n*(n+1)/2 (conjecture); a(n) = n*(n+1)/2 * A300872(n).

Examples

			O.g.f.: A(x) = x + 3*x^2 + 48*x^3 + 1510*x^4 + 71280*x^5 + 4511808*x^6 + 361640832*x^7 + 35516910960*x^8 + 4184770003200*x^9 + ...
where
exp(A(x)) = 1 + x + 7*x^2/2! + 307*x^3/3! + 37537*x^4/4! + 8755561*x^5/5! + 3304572391*x^6/6! + 1847063377867*x^7/7! + 1447456397632897*x^8/8! + ... + A300870(n)*x^n/n! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp( n*(n+1) * A(x) ) begins:
n=1: [(1), (2), 8, 328/3, 9728/3, 2241184/15, 420248704/45, ...];
n=2: [1, (6), (36), 432, 11328, 2470464/5, 150254784/5, ...];
n=3: [1, 12, (108), (1296), 29136, 5776128/5, 335166336/5, ...];
n=4: [1, 20, 260, (10480/3), (209600/3), 7265600/3, 1173400640/9, ...];
n=5: [1, 30, 540, 8640, (166800), (5004000), 241367040, 116509893120/7...];
n=6: [1, 42, 1008, 19656, 396816, (53339328/5), (2240251776/5), ...];
n=7: [1, 56, 1736, 124096/3, 2767184/3, 355355392/15, (38932329856/45), (2180210471936/45), ...]; ...
in which the coefficients in parenthesis are related by
2 = 1*2*(1); 36 = 2*3*(6); 1296 = 3*4*(108); 209600/3 = 4*5*(10480/3); 5004000 = 5*6*(166800); 2240251776/5 = 6*7*(53339328/5); ...
illustrating: [x^n] exp( n*(n+1) * A(x) ) = n*(n+1) * [x^(n-1)] exp( n*(n+1) * A(x) ).
The values A300872(n) = a(n) / (n*(n+1)/2) begin:
[1, 1, 8, 151, 4752, 214848, 12915744, 986580860, 92994888960, ...]
and appear to consist entirely of integers.

Links

Crossrefs

Cf. A300870, A300872, A300591, A296171, A300874.

Programs

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

A295812 G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296170.

Original entry on oeis.org

1, 1, 3, 19, 226, 4259, 110514, 3626207, 143043592, 6567931068, 343278693103, 20092744961109, 1300754163383700, 92223505422990050, 7104166647498916816, 590661172651143976231, 52710327177111760030280, 5024720072707894279118236, 509553454073135435969780828, 54771493019290133717304608756, 6220332385328132888848047735930, 744260531662484056612631555859467
Offset: 1

Views

Author

Paul D. Hanna, Dec 09 2017

Keywords

Comments

E.g.f. G(x) of A296170 satisfies: [x^(n-1)] G(x)^(n^2) = [x^n] G(x)^(n^2) for n>=1.

Examples

			G.f. A(x) = x + x^2 + 3*x^3 + 19*x^4 + 226*x^5 + 4259*x^6 + 110514*x^7 + 3626207*x^8 + 143043592*x^9 + 6567931068*x^10 + 343278693103*x^11 + 20092744961109*x^12 + 1300754163383700*x^13 + 92223505422990050*x^14 + 7104166647498916816*x^15 +...
The series reversion equals the logarithm of the e.g.f. of A296170, which begins:
Series_Reversion(A(x)) = x - x^2 - x^3 - 9*x^4 - 134*x^5 - 2852*x^6 - 79096*x^7 - 2699480*x^8 - 109201844*x^9 - 5100872244*x^10 - 269903909820*x^11 - 15944040740604*x^12 - 1039553309158964*x^13 - 74123498185170292*x^14 - 5736368141560365292*x^15 +...+ A296171(n)*x^n +...

Links

Crossrefs

Cf. A296170, A296171, A295813, A295814.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^2 ); polcoeff(serreverse(log(Ser(A))),n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. is the series reversion of the logarithm of the e.g.f. of A296170.
a(n) ~ c * d^n * n! / n^3, where d = -4/(LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = (2 + LambertW(-2*exp(-2)))^2 * sqrt(-LambertW(-2*exp(-2))*(1 + LambertW(-2*exp(-2)))) / (8*sqrt(2)*Pi) = 0.0350943105... - Vaclav Kotesovec, Dec 22 2017, updated Aug 06 2018

A300619 O.g.f. A(x) satisfies: [x^n] exp( n * A(x) ) = n^3 * [x^(n-1)] exp( n * A(x) ) for n>=1.

Original entry on oeis.org

1, 7, 207, 14226, 1852800, 409408077, 142286748933, 73448832515952, 53835885818473473, 54041298732304775000, 72129250579997923194091, 124900802377559946754633602, 274851919918333747166200590840, 755158633069275870471471631726803, 2551279948230221759814139760682442500
Offset: 1

Views

Author

Paul D. Hanna, Mar 10 2018

Keywords

Comments

O.g.f. equals the logarithm of the e.g.f. of A300618.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = x + 7*x^2 + 207*x^3 + 14226*x^4 + 1852800*x^5 + 409408077*x^6 + 142286748933*x^7 + 73448832515952*x^8 + 53835885818473473*x^9 + ...
where
exp(A(x)) = 1 + x + 15*x^2/2! + 1285*x^3/3! + 347065*x^4/4! + 224232501*x^5/5! + 296201195791*x^6/6! + 719274160258585*x^7/7! + ... + A300618(n)*x^n/n! + ...
such that: [x^n] exp( n * A(x) ) = n^3 * [x^(n-1)] exp( n * A(x) ).

Links

Crossrefs

Cf. A300618, A296171, A300591, A300593, A300595, A300597, A300617.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^3*V[#A-1] - V[#A])/(#A-1) ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))

A300874 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1) * A(x) ) = 2*n * [x^(n-1)] exp( n*(n+1) * A(x) ) for n>=1.

Original entry on oeis.org

1, 1, 6, 78, 1560, 41484, 1361640, 52824144, 2355612192, 118455668960, 6624336466880, 407637626194080, 27374154691010816, 1992569727194556608, 156335075280459423360, 13158244845212096286720, 1183162080050737698802176, 113244610738097834450007552, 11500380596282466998941623296, 1235555832300741998445513374720, 140061215510759508434434106953728
Offset: 1

Views

Author

Paul D. Hanna, Mar 14 2018

Keywords

Comments

Compare to: [x^n] exp( n*(n+1) * x ) = (n+1) * [x^(n-1)] exp( n*(n+1) * x ) for n>=1.
O.g.f. equals the logarithm of the e.g.f. of A300873.
It is conjectured that this sequence consists entirely of integers.

Examples

			O.g.f.: A(x) = x + x^2 + 6*x^3 + 78*x^4 + 1560*x^5 + 41484*x^6 + 1361640*x^7 + 52824144*x^8 + 2355612192*x^9 + 118455668960*x^10 + ...
where
exp(A(x)) = 1 + x + 3*x^2/2! + 43*x^3/3! + 2041*x^4/4! + 197721*x^5/5! + 31094251*x^6/6! + 7086479443*x^7/7! + 2187876597873*x^8/8! + 874871971357681*x^9/9! + ... + A300873(n)*x^n/n! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp( n*(n+1) * A(x) ) begins:
n=1: [(1), (2), 4, 52/3, 560/3, 52304/15, 4048864/45, 914958416/315, ...];
n=2: [1, (6), (24), 108, 864, 67104/5, 1601424/5, 348254352/35, ...];
n=3: [1, 12, (84), (504), 3600, 211968/5, 4273776/5, 860107104/35, ...];
n=4: [1, 20, 220, (5560/3), (44480/3), 438400/3, 20480720/9, 3534944800/63, ...];
n=5: [1, 30, 480, 5580, (55440), (554400), 6991920, 947466000/7, ...];
n=6: [1, 42, 924, 14364, 181440, (10403568/5), (124842816/5), 1922103792/5, ...];
n=7: [1, 56, 1624, 98224/3, 1566992/3, 107909312/15, (4208547616/45), (58919666624/45), ...]; ...
in which the coefficients in parenthesis are related by
2 = 2*1*(1); 24 = 2*2*(6); 504 = 2*3*(84); 44480/3 = 2*4*(5560/3); 554400 = 2*5*(55440); 124842816/5 = 2*6*(10403568/5); ...
illustrating: [x^n] exp( n*(n+1) * A(x) ) = 2*n * [x^(n-1)] exp( n*(n+1) * A(x) ).

Links

Crossrefs

Cf. A300873, A300871, A300591, A296171.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)*(#A))); A[#A] = (2*(#A-1)*V[#A-1] - V[#A])/(#A-1)/(#A) ); polcoeff( log(Ser(A)), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

a(n) ~ c * d^n * n! / n^3, where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.75891265... - Vaclav Kotesovec, Mar 14 2018

A300615 O.g.f. A(x) satisfies: [x^n] exp( n^5 * A(x) ) = n^5 * [x^(n-1)] exp( n^5 * A(x) ) for n>=1.

Original entry on oeis.org

1, 16, 19683, 142475264, 3436799053125, 212148041589128016, 28458158819417861315152, 7380230750280159370894934016, 3385049575573746853297963891959753, 2561548157856026756893458765378989150000, 3026444829408778969259555715061437179090541565, 5340113530831632053993990154143996936096662034267136
Offset: 1

Views

Author

Paul D. Hanna, Mar 10 2018

Keywords

Comments

Compare to: [x^n] exp( n^5 * x ) = n^4 * [x^(n-1)] exp( n^5 * x ) for n>=1.

Examples

			O.g.f.: A(x) = x + 16*x^2 + 19683*x^3 + 142475264*x^4 + 3436799053125*x^5 + 212148041589128016*x^6 + 28458158819417861315152*x^7 + ...
where
exp(A(x)) = 1 + x + 33*x^2/2! + 118195*x^3/3! + 3419881993*x^4/4! + 412433022394701*x^5/5! + 152749066271797582081*x^6/6! + ... + A300614(n)*x^n/n! + ...
such that: [x^n] exp( n^5 * A(x) ) = n^5 * [x^(n-1)] exp( n^5 * A(x) ).

Links

Crossrefs

Cf. A300614, A296171, A300591, A300593, A300595, A300597.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^5)); A[#A] = ((#A-1)^5*V[#A-1] - V[#A])/(#A-1)^5 ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))

Formula

O.g.f. equals the logarithm of the e.g.f. of A300614.
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