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 1-6 of 6 results.

A054842 If n = a + 10 * b + 100 * c + 1000 * d + ... then a(n) = (2^a) * (3^b) * (5^c) * (7^d) * ...

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 27, 54, 108, 216, 432, 864, 1728, 3456, 6912, 13824, 81, 162, 324, 648, 1296, 2592, 5184, 10368, 20736, 41472, 243, 486, 972, 1944
Offset: 0

Views

Author

Henry Bottomley, Apr 11 2000

Keywords

Comments

a((10^k-1)/9) = Primorial(k)= A061509((10^k-1)/9). This is a rearrangement of whole numbers. a(m) = a(n) iff m = n. (Unlike A061509, in which a(n) = a(n*10^k).) - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 14 2003
Part of the previous comment is incorrect: as a set, this sequence consists of numbers n such that the largest exponent appearing in the prime factorization of n is 9. So this cannot be a rearrangement (or permutation) of the natural numbers. - Tom Edgar, Oct 20 2015

Examples

			a(15)=96 because 3^1 * 2^5 = 3*32 = 96.

Links

Crossrefs

Cf. A054841, A069877, A085840.
Cf. analogous sequences for other bases: A019565 (base 2), A101278 (base 3), A101942 (base 4), A101943 (base 5), A276076 (factorial base), A276086 (primorial base).

Programs

  • Haskell
    a054842 = f a000040_list 1 where
   f _      y 0 = y
   f (p:ps) y x = f ps (y * p ^ d) x'  where (x', d) = divMod x 10
-- Reinhard Zumkeller, Aug 03 2015
  • Mathematica
    A054842[n_] := Times @@ (Prime[Range[Length[#], 1, -1]]^#) & [IntegerDigits[n]];
Array[A054842, 100, 0] (* Paolo Xausa, Nov 25 2024 *)
  • PARI
    a(n)= my(d=Vecrev(digits(n))); factorback(primes(#d), d); \\ Ruud H.G. van Tol, Nov 28 2024

Formula

a(n) = f(n, 1, 1) with f(x, y, z) = if x > 0 then f(floor(x/10), y*prime(z)^(x mod 10), z+1) else y. - Reinhard Zumkeller, Mar 13 2010

A285043 Expansion of cosh(3*arctanh(2*sqrt(x))).

Original entry on oeis.org

1, 18, 102, 500, 2310, 10332, 45276, 195624, 836550, 3549260, 14965236, 62783448, 262303132, 1092063000, 4533175800, 18769219920, 77539370310, 319704052140, 1315894618500, 5407825361400, 22193291140020
Offset: 0

Views

Author

Peter Bala, Apr 09 2017

Keywords

Comments

Note that the function cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. See, for example, A010006.
In A285043 through A285046 we consider sequences with o.g.f. cosh((2*n+1)*arctanh(2*sqrt(x))) for n = 1, 2, 3 and 4. For n = 0 we get the central binomial coefficients A000984.

Crossrefs

Cf. A000984, A010006, A085840, A285044, A285045, A285046.

Programs

  • Maple
    seq((8*n + 1)*binomial(2*n,n), n = 0..20);
  • Mathematica
    CoefficientList[Series[Cosh[3*ArcTanh[2*Sqrt[x]]], {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 10 2017 *)
  • PARI
    my(x='x + O('x^30)); Vec((1 + 12*x)/(1 - 4*x)^(3/2)) \\ Indranil Ghosh, Apr 10 2017

Formula

a(n) = (8*n + 1)*binomial(2*n,n).
O.g.f. cosh(3*arctanh(2*sqrt(x))) = (1 + 12*x)/(1 - 4*x)^(3/2) = 1 + 18*x + 102*x^2 + 500*x^3 + ....
D-finite with recurrence: n*a(n) +2*(4*n-13)*a(n-1) +24*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jan 22 2020

A285046 Expansion of cosh(9*arctanh(2*sqrt(x))).

Original entry on oeis.org

1, 162, 4806, 71892, 758214, 6506172, 48783900, 332715240, 2115552582, 12745645484, 73577414196, 410265444888, 2222886926364, 11756568121560, 60911288332920, 310024235290320, 1553692427724870
Offset: 0

Views

Author

Peter Bala, Apr 10 2017

Keywords

Comments

Note that the function cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. See, for example, A010006.
In A285043 through A285046 we consider sequences with o.g.f. cosh((2*n + 1)*arctanh(2*sqrt(x))) for n = 1, 2, 3 and 4: n = 0 gives the central binomial coefficients A000984.

Crossrefs

Cf. A000984, A010006, A085840, A285043, A285044, A285045.

Programs

  • Maple
    seq(1/105*(4096*n^4 + 512*n^3 + 3392*n^2 + 400*n + 105)*binomial(2*n,n), n = 0..20);
  • PARI
    x='x + O('x^30); print(Vec((1 + 144*x + 2016*x^2 + 5376*x^3 + 2304*x^4)/(1 - 4*x)^(9/2))) \\ Indranil Ghosh, Apr 10 2017

Formula

a(n) = 1/105*(4096*n^4 + 512*n^3 + 3392*n^2 + 400*n + 105)*binomial(2*n,n).
O.g.f. cosh(9*arctanh(2*sqrt(x))) = (1 + 144*x + 2016*x^2 + 5376*x^3 + 2304x^4)/(1 - 4*x)^(9/2) = 1 + 162*x + 4806*x^2 + 71892*x^3 + ....
Note that the zeros of the polynomial 1 + 144*x^2 + 2016*x^4 + 5376*x^6 + 2304*x^8 = 1/2*((1 + 2*x)^9 + (1 - 2*x)^9), are given by 1/2*cot(k*Pi/9)*i for 1 <= k <= 8. See A085840.
O.g.f. for the sequence with interpolated zeros: 1/2*( ((1 + 2*x)/(1 - 2*x))^(9/2) + ((1 - 2*x)/(1 + 2*x))^(9/2) ) = 1 + 162*x^2 + 4806*x^4 + 71892*x^6 + ....

A285045 Expansion of cosh(7*arctanh(2*sqrt(x))).

Original entry on oeis.org

1, 98, 1862, 19796, 160454, 1114428, 7008540, 41132520, 229435206, 1230873644, 6403088692, 32488200472, 161473267228, 788758622680, 3796375603320, 18040943163600, 84786596572230, 394599588033420, 1820669979129540, 8335975464699960
Offset: 0

Views

Author

Peter Bala, Apr 10 2017

Keywords

Comments

Note that the function cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. See, for example, A010006.
In A285043 through A285046 we consider sequences with o.g.f. cosh((2*n+1)*arctanh(2*sqrt(x))) for n = 1, 2, 3 and 4: n = 0 gives the central binomial coefficients A000984.

Crossrefs

Cf. A000984, A010006, A085840, A285043, A285044, A285046.

Programs

  • Maple
    seq(1/15*(512*n^3 + 64*n^2 + 144*n + 15)*binomial(2*n,n), n = 0..20);
  • Mathematica
    CoefficientList[Series[Cosh[7*ArcTanh[2Sqrt[x]]],{x,0,20}],x] (* Harvey P. Dale, Jun 07 2024 *)

Formula

a(n) = 1/15*(512*n^3 + 64*n^2 + 144*n + 15)*binomial(2*n,n).
O.g.f. cosh(7*arctanh(2*sqrt(x))) = (1 + 84*x + 560*x^2 + 448*x^3)/(1 - 4*x)^(7/2) = 1 + 98*x + 1862*x^2 + 19796*x^3 + ....
Note that the zeros of the polynomial 1 + 84*x^2 + 560*x^4 + 448*x^6 = 1/2*((1 + 2*x)^7 + (1 - 2*x)^7), are given by 1/2*cot(k*Pi/7)*i for 1 <= k <= 6. See A085840.
O.g.f. for the sequence with interpolated zeros: 1/2*( ((1 + 2*x)/(1 - 2*x))^(7/2) + ((1 - 2*x)/(1 + 2*x))^(7/2) ) = 1 + 98*x^2 + 1862*x^4 + 19796*x^6 + ....
D-finite with recurrence: n*(2*n-1)*a(n) +2*(-8*n^2+16*n-57)*a(n-1) +16*(2*n-3)*(n-2)*a(n-2)=0. - R. J. Mathar, Jan 22 2020

A285044 Expansion of cosh(5*arctanh(2*sqrt(x))).

Original entry on oeis.org

1, 50, 550, 4020, 24710, 138012, 725340, 3655080, 17859270, 85230860, 399257716, 1842353240, 8396404380, 37868584600, 169278679800, 750923914320, 3308947546950, 14495583969900, 63172016823300, 274031830241400, 1183780040663220
Offset: 0

Views

Author

Peter Bala, Apr 10 2017

Keywords

Comments

Note that the function cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. See, for example, A010006.
In A285043 through A285046 we consider sequences with o.g.f. cosh((2*n+1)*arctanh(2*sqrt(x))) for n = 1, 2, 3 and 4: n = 0 gives the central binomial coefficients A000984.

Crossrefs

Cf. A000984, A010006, A085840, A285043, A285045, A285046.

Programs

  • Maple
    seq(1/3*(64*n^2 + 8*n + 3)*binomial(2*n,n), n = 0..20);

Formula

a(n) = 1/3*(64*n^2 + 8*n + 3)*binomial(2*n,n).
O.g.f. cosh(5*arctanh(2*sqrt(x))) = (1 + 40*x + 80*x^2)/(1 - 4*x)^(5/2) = 1 + 50*x + 550*x^2 + 4020*x^3 + ....
Note that the zeros of the polynomial 1 + 40*x^2 + 80*x^4 = 1/2*((1 + 2*x)^5 + (1 - 2*x)^5), are given by 1/2*cot(k*Pi/5)*i for 1 <= k <= 4. See A085840.
O.g.f. for the sequence with interpolated zeros: 1/2*( ((1 + 2*x)/(1 - 2*x))^(5/2) + ((1 - 2*x)/(1 + 2*x))^(5/2) ) = 1 + 50*x^2 + 550*x^4 + 4020*x^6 + ....

A085841 Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).

Original entry on oeis.org

1, 3, 4, 5, 40, 16, 7, 140, 336, 64, 9, 336, 2016, 2304, 256, 11, 660, 7392, 21120, 14080, 1024, 13, 1144, 20592, 109824, 183040, 79872, 4096, 15, 1820, 48048, 411840, 1281280, 1397760, 430080, 16384
Offset: 0

Views

Author

Gary W. Adamson, Jul 05 2003

Keywords

Comments

Row #n has the unsigned coefficients of a polynomial whose roots are 2 cot (Pi k / (2n+1)) for k=1..2n.
Polynomial of row #n = Sum_{m=0..n} (-1)^m*T(n,m)*x^(2n-2m).

Examples

			1
3x^2 - 4
5x^4 - 40x^2 + 16
7x^6 - 140x^4 + 336x^2 - 64
9x^8 - 336x^6 + 2016x^4 - 2304x^2 + 256
11x^10 - 660x^8 + 7392x^6 - 21120x^4 + 14080x^2 - 1024
Polynomial #4 has eight roots: 2 cot (Pi k / 9) for k=1..8.

Links

Crossrefs

Cf. A085840.

Programs

  • PARI
    T(n,m) = 4^m*(2*n+1)!/((2*n-2*m)!*(2*m+1)!);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 18 2018

Extensions

Edited by Don Reble, Nov 13 2005
Showing 1-6 of 6 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.