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.

A304857 a(n) = A304400(n) / n^2 for n >= 1.

Original entry on oeis.org

1, 2, 17, 296, 8205, 322412, 16861408, 1129044208, 94262593881, 9617095872812, 1179417144559337, 171422055924510232, 29164680268271517644, 5744954740594875376712, 1297633915706481674014856, 333220154688593666676579264, 96542160783658837442386115393, 31344432027934429150200553978812
Offset: 1

Views

Author

Paul D. Hanna, May 25 2018

Keywords

Links

Crossrefs

Cf. A304400.

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

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

A304402 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x) ) / A(x) = 0 for n > 0.

Original entry on oeis.org

1, 1, 9, 179, 5661, 249424, 14337039, 1035838044, 91867414241, 9833503227827, 1253246430314670, 187948018130914066, 32818034910964227439, 6608081830970361618546, 1520982783352578794866344, 397027611766464517915252056, 116698001659938095895315068553, 38375694701199964362412343063161
Offset: 0

Views

Author

Paul D. Hanna, May 25 2018

Keywords

Comments

Note: [x^n] exp( n * x*G(x) ) / G(x) = 0 for n>0 when G(x) is the g.f. of A088716.
It is remarkable that this sequence should consist entirely of integers.
What is the limit A304402(n) / A304400(n) ? Seems to be near 1.51...
A304402(n) / A304400(n) tends to 1.522998920075488836991600223419379... - Vaclav Kotesovec, Oct 06 2020

Examples

			O.g.f.: A(x) = 1 + x + 9*x^2 + 179*x^3 + 5661*x^4 + 249424*x^5 + 14337039*x^6 + 1035838044*x^7 + 91867414241*x^8 + 9833503227827*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2 * x*A(x) ) / A(x) begins:
n=0: [1, -1, -16, -972, -125952, -28275000, -9885939840, ...];
n=1: [1, 0, -15, -968, -125835, -28263864, -9883855835, ...];
n=2: [1, 3, 0, -860, -123456, -28073976, -9850185728, ...];
n=3: [1, 8, 65, 0, -104811, -26970576, -9680119083, ...];
n=4: [1, 15, 240, 3892, 0, -21937464, -9078485120, ...];
n=5: [1, 24, 609, 16528, 457173, 0, -7077136715, ...];
n=6: [1, 35, 1280, 49572, 2066880, 89033736, 0, ...];
n=7: [1, 48, 2385, 123880, 6839349, 411165624, 26124539077, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^2 * x*A(x) ) / A(x) = 0 for n > 0.
Terms along the secondary diagonal in the above table are divisible by the odd numbers: [1, 3/3, 65/5, 3892/7, 457173/9, 89033736/11, 26124539077/13, ...] = [1, 1, 13, 556, 50797, 8093976, 2009579929, ...].
RELATED SERIES.
exp( x*A(x) ) = 1 + x + 3*x^2/2! + 61*x^3/3! + 4537*x^4/4! + 702501*x^5/5! + 183891571*x^6/6! + 73567995313*x^7/7! + 42361186187601*x^8/8! + ...
The arithmetic inverse of the o.g.f. begins:
1/A(x) = 1 - x - 8*x^2 - 162*x^3 - 5248*x^4 - 235625*x^5 - 13730472*x^6 - 1001798042*x^7 - 89479215104*x^8 - 9627430506669*x^9 + ...

Links

Crossrefs

Cf. A304400, A088716.

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

Formula

a(n) ~ c * n!^2 * n^2, where c = 1.18365083976367345437640389636650727... - Vaclav Kotesovec, Oct 06 2020

A304401 O.g.f. A(x) satisfies: [x^n] exp( n^3 * x*A(x) ) * (2 - A(x)) = 0 for n > 0.

Original entry on oeis.org

1, 1, 32, 3618, 845824, 332389375, 196888240512, 164288952970296, 184344892426059776, 268830705445490506509, 496348897291481486672000, 1136486246811467501138927540, 3173564392477075053313688696832, 10660730426979559461604460186833401, 42595326050479099018430338636152049280, 200526023793925980859314834103239034380000
Offset: 0

Views

Author

Paul D. Hanna, May 25 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.
Note: a(n) is divisible by n^3 for n >= 1.

Examples

			O.g.f.: A(x) = 1 + x + 32*x^2 + 3618*x^3 + 845824*x^4 + 332389375*x^5 + 196888240512*x^6 + 164288952970296*x^7 + 184344892426059776*x^8 + ...
ILLUSTRATION OF SEFINITION.
The table of coefficients of x^k/k! in exp( n^3 * x*A(x) ) * (2 - A(x)) begins:
n=0: [1, -1, -64, -21708, -20299776, -39886725000, ...];
n=1: [1, 0, -63, -21710, -20300931, -39887501724, ...];
n=2: [1, 7, 0, -21052, -20280064, -39880261512, ...];
n=3: [1, 26, 665, 0, -19381155, -39710564418, ...];
n=4: [1, 63, 4032, 252340, 0, -37416032136, ...];
n=5: [1, 124, 15561, 1977542, 245086349, 0, ...];
n=6: [1, 215, 46592, 10194660, 2254128384, 485581472376, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^3 * x*A(x) ) * (2 - A(x)) = 0 for n > 0.
Terms along the secondary diagonal in the above table are divisible by the differences of cubes: [1, 7/7, 665/19, 252340/37, 245086349/61, 485581472376/91, ...] = [1, 1, 35, 6820, 4017809, 5336060136, ...].
RELATED SERIES.
exp( x*A(x) ) = 1 + x + 3*x^2/2! + 199*x^3/3! + 87625*x^4/4! + 101938881*x^5/5! + 239933646571*x^6/6! + 993998976594583*x^7/7! + 6632090620377452049*x^8/8! + ...
Note that the factorial series
F(x) = 1 + x + 2!*x^2 + 3!*x^3 + 4!*x^4 + 5!*x^5 + ... + n!*x^n + ...
satisfies [x^n] exp( n*x*F(x) ) * (2 - F(x)) = 0 for n > 0.

Crossrefs

Cf. A304400.

Programs

  • PARI
    {a(n) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( (m-1)^3 * x * Ser(A) ) * (2 - Ser(A)) )[m] );A[n+1]}
for(n=0,20, print1(a(n),", "))
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.