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 1-3 of 3 results.

A305116 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x) ) * (n + 1 - A(x)) = 0 for n >= 0, where A(0) = 1.

Original entry on oeis.org

1, 1, 20, 918, 80032, 12042925, 2930093028, 1091180685420, 593430683068672, 453081063936151719, 469964400518950271900, 644367335619103754943450, 1141157288474505534959353440, 2559472926372019471694595185328, 7148083254588411836230809315647744, 24494543545202626717977721555958466300, 101668844348061438731562868186881235350528
Offset: 0

Views

Author

Paul D. Hanna, May 26 2018

Keywords

Comments

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
(1) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0,
(2) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + x + 20*x^2 + 918*x^3 + 80032*x^4 + 12042925*x^5 + 2930093028*x^6 + 1091180685420*x^7 + 593430683068672*x^8 + 453081063936151719*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2 * x*A(x) ) * (n + 1 - A(x)) begins:
n=0: [0, -1, -40, -5508, -1920768, -1445151000, -2109666980160, ...];
n=1: [1, 0, -39, -5510, -1921491, -1445365884, -2109780457715, ...];
n=2: [2, 7, 0, -4780, -1823168, -1405023192, -2074130121472, ...];
n=3: [3, 26, 239, 0, -1391649, -1249241538, -1942417653741, ...];
n=4: [4, 63, 1080, 21916, 0, -860673816, -1637736990272, ...];
n=5: [5, 124, 3285, 101342, 4459057, 0, -1050171876535, ...];
n=6: [6, 215, 8096, 338580, 18744384, 1958675496, 0, ...];
n=7: [7, 342, 17355, 946660, 61910307, 6852230778, 1865443733743, 0, ...]; ...
in which the coefficients of x^n in row n form a diagonal of zeros.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 127*x^3/3! + 22537*x^4/4! + 9717681*x^5/5! + 8729681611*x^6/6! + 14829069291583*x^7/7! + 44115361026430737*x^8/8! + ...

Links

Crossrefs

Cf. A305110, A305114, A305115, A304400, A317337.

Programs

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

Formula

a(n) ~ c * n!^3, where c = 13.46489329292094724950380929883219... - Vaclav Kotesovec, Oct 06 2020

A305115 O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n^2 + 1 - A(x)) = 0 for n >= 0.

Original entry on oeis.org

1, 1, 14, 450, 31144, 4041775, 890769366, 309205147860, 159530833094816, 116905524905145753, 117339344873068964150, 156605173710780053035502, 271173392660354548224099528, 596723380510396302812115056135, 1639486267597614501043345413095854, 5538914776834654404464150449671117000, 22706307619073102796968257487359193429120
Offset: 0

Views

Author

Paul D. Hanna, May 26 2018

Keywords

Comments

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
(1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,
(2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + x + 14*x^2 + 450*x^3 + 31144*x^4 + 4041775*x^5 + 890769366*x^6 + 309205147860*x^7 + 159530833094816*x^8 + 116905524905145753*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n^2 + 1 - A(x)) begins:
n=0: [0, -1, -28, -2700, -747456, -485013000, -641353943520, ...];
n=1: [1, 0, -27, -2702, -747963, -485118684, -641396951615, ...];
n=2: [4, 7, 0, -2092, -678784, -462055752, -623679177536, ...];
n=3: [9, 26, 101, 0, -460275, -391250658, -569892209247, ...];
n=4: [16, 63, 348, 4828, 0, -246538056, -461135488928, ...];
n=5: [25, 124, 837, 14150, 810509, 0, -277891671695, ...];
n=6: [36, 215, 1688, 30348, 2099712, 378224376, 0, ...];
n=7: [49, 342, 3045, 56548, 4020741, 920163738, 393372598609, 0, ...]; ...
in which the coefficients of x^n in row n form a diagonal of zeros.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 91*x^3/3! + 11161*x^4/4! + 3793881*x^5/5! + 2933070331*x^6/6! + 4510118566003*x^7/7! + 12503335235913201*x^8/8! + ...

Links

Crossrefs

Cf. A305110, A305114, A305116, A317337.

Programs

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

Formula

a(n) ~ c * n!^3, where c = 2.49393609789981559563078907122202821077556480458411... - Vaclav Kotesovec, Oct 06 2020

A317338 O.g.f. A(x) satisfies: [x^n] exp( n*x*A(x) ) * (n+1 - n*A(x)) = 0 for n >= 1.

Original entry on oeis.org

1, 1, 0, -3, -5, 10, 58, 23, -557, -1421, 4094, 28316, -52, -449150, -970286, 5908939, 31046627, -49583353, -750617284, -544416915, 15819383275, 46795708732, -288245326872, -1808819140124, 3784215933076, 57664747490276, 14416027504376, -1664155475303224, -3937904190952656, 43893853942734810, 219165998056699650
Offset: 0

Views

Author

Paul D. Hanna, Aug 01 2018

Keywords

Comments

Compare: the factorial series F(x) = Sum_{n>=0} n!*x^n satisfies
(1) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n >= 1,
(2) [x^n] exp( n*x*F(x) ) * (2 - F(x)) = 0 for n >= 1,
(3) [x^n] exp( n^2*x*F(x) ) * (n + 1 - n*F(x)) = 0 for n >= 1.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + x - 3*x^3 - 5*x^4 + 10*x^5 + 58*x^6 + 23*x^7 - 557*x^8 - 1421*x^9 + 4094*x^10 + 28316*x^11 - 52*x^12 - 449150*x^13 - 970286*x^14 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n*x*A(x) ) * (n+1 - n*A(x)) begins:
n=1: [1, 0, 1, 16, 117, -704, -35075, -200304, 17660041, ...];
n=2: [1, 0, 0, 20, 288, 912, -51200, -888480, 19165440, ...];
n=3: [1, 0, -3, 0, 333, 3888, -27135, -1471824, 4665465, ...];
n=4: [1, 0, -8, -56, 0, 5344, 33280, -1317312, -15647744, ...];
n=5: [1, 0, -15, -160, -1035, 0, 81325, -180000, -25008375, ...];
n=6: [1, 0, -24, -324, -3168, -20304, 0, 1156896, -10209024, ...];
n=7: [1, 0, -35, -560, -6867, -67088, -422975, 0, 19205305, ...];
n=8: [1, 0, -48, -880, -12672, -155712, -1525760, -9408384, 0, ...];
n=9: [1, 0, -63, -1296, -21195, -305856, -3806595, -37346832, -230393079, 0, ...]; ...
in which the coefficients of x^n in row n form a diagonal of zeros.

Links

Crossrefs

Cf. A317337, A305110, A305114, A305115, A305116.

Programs

  • PARI
    {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*x*Ser(A) ) * (m - (m-1)*Ser(A)) )[m]/(m-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.

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