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 1-4 of 4 results.

A300872 a(n) = A300871(n) / (n*(n+1)/2).

Original entry on oeis.org

1, 1, 8, 151, 4752, 214848, 12915744, 986580860, 92994888960, 10595684332288, 1436363905680384, 228679178713630208, 42284602089642237952, 8992606241049735405568, 2180532527491138011131904, 598191016068264518151780096, 184370870332464252513762869248, 63445183762362816656030378164224, 24238363163428954774170892697075712
Offset: 1

Views

Author

Paul D. Hanna, Mar 14 2018

Keywords

Comments

It is conjectured that this sequence consists entirely of integers.
O.g.f. G(x) of A300871 satisfies: [x^n] exp( n*(n+1) * G(x) ) = n*(n+1) * [x^(n-1)] exp( n*(n+1) * G(x) ) for n>=1.

Links

Crossrefs

Cf. A300871.

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) ); (1/(n*(n+1)/2))*polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))

A300591 O.g.f. A(x) satisfies: [x^n] exp( n^2 * A(x) ) = n^2 * [x^(n-1)] exp( n^2 * A(x) ) for n>=1.

Original entry on oeis.org

1, 2, 27, 736, 30525, 1715454, 123198985, 10931897664, 1172808994833, 149774206572050, 22487782439633786, 3927856758905547936, 790620718368726490063, 181836026214536919343314, 47416473117145116482171400, 13920906749656695367066255360, 4572270908185359745686931830057, 1670388578072378805032472463218378, 675225859431899136993903503004997481, 300576566118865697499246162737030656800
Offset: 1

Views

Author

Paul D. Hanna, Mar 09 2018

Keywords

Comments

Compare to: [x^n] exp( n^2 * x ) = n * [x^(n-1)] exp( n^2 * x ) for n>=1.
It is conjectured that this sequence consists entirely of integers.
a(n) is divisible by n (conjecture): A300598(n) = a(n)/n for n>=1.

Examples

			O.g.f.: A(x) = x + 2*x^2 + 27*x^3 + 736*x^4 + 30525*x^5 + 1715454*x^6 + 123198985*x^7 + 10931897664*x^8 + 1172808994833*x^9 + 149774206572050*x^10 + ...
where
exp(A(x)) = 1 + x + 5*x^2/2! + 175*x^3/3! + 18385*x^4/4! + 3759701*x^5/5! + 1258735981*x^6/6! + 630063839035*x^7/7! + 445962163492385*x^8/8! + ... + A300590(n)*x^n/n! + ...
such that: [x^n] exp( n^2 * A(x) ) = n^2 * [x^(n-1)] exp( n^2 * A(x) ).

Links

Crossrefs

Cf. A300590, A300598, A300871, A296171, A300593, A300595.

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] = ((#A-1)^2*V[#A-1] - V[#A])/(#A-1)^2 ); polcoeff( log(Ser(A)), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

O.g.f. equals the logarithm of the e.g.f. of A300590.
a(n) ~ c * n!^2 * n^2, where c = 0.1354708370957778563796... - Vaclav Kotesovec, Oct 13 2020

A300870 E.g.f. A(x) satisfies: [x^n] A(x)^(n*(n+1)) = n*(n+1) * [x^(n-1)] A(x)^(n*(n+1)) for n>=1.

Original entry on oeis.org

1, 1, 7, 307, 37537, 8755561, 3304572391, 1847063377867, 1447456397632897, 1532041772833285777, 2130468278450240803591, 3808068399270998260188451, 8590473242021318921848038817, 24074336129439663228349612217977, 82657249526888437632759608331784807, 343425012928825298349935150449843384891, 1707701025594135213863151839769061397729281
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2018

Keywords

Comments

Compare e.g.f. to: [x^n] exp(x)^(n*(n+1)) = (n+1) * [x^(n-1)] exp(x)^(n*(n+1)) for n>=1.

Examples

			E.g.f.: 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! + 1532041772833285777*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n*(n+1)) 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 that: [x^n] A(x)^(n*(n+1)) = n*(n+1) * [x^(n-1)] A(x)^(n*(n+1)).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(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 + ... + A300871(n)*x^n + ...

Links

Crossrefs

Cf. A300871, A300590, A296170, A182962.

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) ); EGF=Ser(A); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) ~ c * n!^3 * n^3, where c = 0.044039511494832369374... - Vaclav Kotesovec, Oct 14 2020

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
Showing 1-4 of 4 results.

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