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 1-10 of 81 results. Next

A143475 Numerator of the coefficient of z^(2n) in the Stirling-like asymptotic expansion of the hyperfactorial function A002109.

Original entry on oeis.org

1, 1, -1433, 1550887, -365236274341, 31170363588856607, -2626723351027654662151, 127061942835077684151157039, -5696145248370283185291966600124423, 254326794362835881966596504823903633657, -33203124408022060010631772664020406983485604379
Offset: 0

Views

Author

Eric W. Weisstein, Aug 19 2008

Keywords

Examples

			(Glaisher*(1 - 1433/(7257600*z^4) + 1/(720*z^2))*z^(1/12 + (z*(1 + z))/2))/e^(z^2/4).
From _Seiichi Manyama_, Aug 31 2018: (Start)
c_1 = -1/2 * (B_4*c_0/(3*4)) = 1/720, so a(1) = 1.
c_2 = -1/4 * (B_6*c_0/(5*6) + B_4*c_1/(3*4)) = -1433/7257600, so a(2) = -1433. (End)

References

  • Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Links

Crossrefs

Cf. A002109, A143476.

Formula

From Seiichi Manyama, Aug 31 2018: (Start)
Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (-1/(2*n)) * Sum_{k=0..n-1} B_{2*n-2*k+2}*c_k/((2*n-2*k+1)*(2*n-2*k+2)) for n > 0.
a(n) is the numerator of c_n. (End)

Extensions

More terms from Seiichi Manyama, Aug 31 2018

A143476 Denominator of the coefficient of z^(2n) in the Stirling-like asymptotic expansion of the hyperfactorial function A002109.

Original entry on oeis.org

1, 720, 7257600, 15676416000, 3476402012160000, 162695614169088000000, 4919915372473221120000000, 60219764159072226508800000000, 507464726196802564122476544000000000, 3288371425755280615513648005120000000000
Offset: 0

Views

Author

Eric W. Weisstein, Aug 19 2008

Keywords

Comments

In Glaisher (1878) equation (2) is "1^1.2^2.3^3 ... n^n = A n^(n^2/2 + n/2 + 1/12) e^(-n^4/4) (1 + 1/(720n^2) - 1433/(7257600n^4) + &c.)" - Michael Somos, Jun 24 2012

Examples

			(Glaisher*(1 - 1433/(7257600*z^4) + 1/(720*z^2))*z^(1/12 + (z*(1 + z))/2))/e^(z^2/4).
From _Seiichi Manyama_, Aug 31 2018: (Start)
c_1 = -1/2 * (B_4*c_0/(3*4)) = 1/720, so a(1) = 720.
c_2 = -1/4 * (B_6*c_0/(5*6) + B_4*c_1/(3*4)) = -1433/7257600, so a(2) = 7257600. (End)

References

  • Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
  • J. W. L. Glaisher, On The Product 1^1.2^2.3^3 ... n^n, Messenger of Mathematics, 7 (1878), pp. 43-47, see p. 43 eq. (2)

Links

Crossrefs

Cf. A002109, A143475.

Formula

From Seiichi Manyama, Aug 31 2018: (Start)
Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (-1/(2*n)) * Sum_{k=0..n-1} B_{2*n-2*k+2}*c_k/((2*n-2*k+1)*(2*n-2*k+2)) for n > 0.
a(n) is the denominator of c_n. (End)

A249152 Exponent of 2 in the hyperfactorials: a(n) = A007814(A002109(n)).

Original entry on oeis.org

0, 0, 2, 2, 10, 10, 16, 16, 40, 40, 50, 50, 74, 74, 88, 88, 152, 152, 170, 170, 210, 210, 232, 232, 304, 304, 330, 330, 386, 386, 416, 416, 576, 576, 610, 610, 682, 682, 720, 720, 840, 840, 882, 882, 970, 970, 1016, 1016, 1208, 1208, 1258, 1258, 1362, 1362, 1416, 1416, 1584, 1584, 1642, 1642
Offset: 0

Views

Author

Antti Karttunen, Oct 25 2014

Keywords

Comments

This is the function ord_2(D*_n) listed in the leftmost column of Table 7.1 in Lagarias & Mehta 2014 paper (on page 19).

Links

Crossrefs

Bisection: A249153.
Cf. A174605, A187059, A002109, A007814, A091512, A143157.
Cf. A133457 (binary exponents).

Programs

  • Magma
    [0] cat [&+[i*Valuation(i, 2):i in [1..n]]:n in [1..60]]; // Marius A. Burtea, Oct 18 2019
  • Maple
    with(padic): seq(add(i*ordp(i, 2), i=1..n), n=0..60); # Ridouane Oudra, Oct 17 2019
  • Mathematica
    Table[i=0;Hyperfactorial@n//.x_/;EvenQ@x:>(i++;x/2);i,{n,0,60}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)
  • PARI
    a(n) = sum(i=1, n, i*valuation(i, 2)); \\ Michel Marcus, Sep 14 2021
  • PARI
    a(n) = my(v=binary(n),t=0); forstep(j=#v,1,-1, if(v[j],v[j]=t--,t++)); (n^2 + fromdigits(v,2))>>1; \\ Kevin Ryde, Nov 03 2021
  • Python
    def A249152(n): return sum(i*(~i&i-1).bit_length() for i in range(2,n+1,2)) # Chai Wah Wu, Jul 11 2022

Formula

a(n) = 2 * A143157(floor(n/2)).
a(n) = A174605(n) + A187059(n). [Lagarias and Mehta theorem 4.1 for p=2]
a(n) = Sum_{i=1..n} i*v_2(i), where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i. - Ridouane Oudra, Oct 17 2019
a(n) ~ (n^2+2n)/2 as n -> infinity. - Luca Onnis, Oct 17 2021
a(n) ~ ((A011371(n))^2)/2 as n -> infinity. - Luca Onnis, Nov 02 2021
From Kevin Ryde, Nov 03 2021: (Start)
a(2n) = a(2n+1) = 2*a(n) + n*(n+1).
a(n) = ( n^2 + Sum_{j=1..k} (e[j]-2*j+1) * 2^e[j] )/2, where binary expansion n = 2^e[1] + ... + 2^e[k] with ascending exponents e[1] < e[2] < ... < e[k] (A133457).
(End)
a(n) = Sum_{j=1..floor(log_2(n))} j*2^j*round(n/2^(j+1))^2, for n>=1. - Ridouane Oudra, Oct 01 2022

A240993 A000142 (n+1) * A002109(n), a product of factorials and hyperfactorials.

Original entry on oeis.org

1, 2, 24, 2592, 3317760, 62208000000, 20316635136000000, 133852981198454784000000, 20211123400293732996612096000000, 78302033109811407811828935756349440000000, 8613223642079254859301182933198438400000000000000000
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 31 2014

Keywords

Comments

a(n+1) / a(n) = A055897(n+2);
row products of the triangle A245334.

Links

Crossrefs

Cf. A000142, A002109, A245334, A055897, A245334, A111063, A074962.

Programs

  • Haskell
    a240993 n = a000142 (n + 1) * a002109 n
  • Mathematica
    Table[(n+1)!*Hyperfactorial[n], {n, 0, 10}] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[(n+1)*(n!)^(n+1)/BarnesG[n+1], {n, 0, 10}] (* Vaclav Kotesovec, Nov 14 2014 *)

Formula

a(n) ~ A * sqrt(2*Pi) * n^(n^2/2+3*n/2+19/12) / exp(n*(n+4)/4), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 14 2014

A249153 Exponent of 2 in the hyperfactorial of 2n: a(n) = A007814(A002109(2n)).

Original entry on oeis.org

0, 2, 10, 16, 40, 50, 74, 88, 152, 170, 210, 232, 304, 330, 386, 416, 576, 610, 682, 720, 840, 882, 970, 1016, 1208, 1258, 1362, 1416, 1584, 1642, 1762, 1824, 2208, 2274, 2410, 2480, 2696, 2770, 2922, 3000, 3320, 3402, 3570, 3656, 3920, 4010, 4194, 4288, 4768, 4866, 5066, 5168, 5480, 5586, 5802, 5912
Offset: 0

Views

Author

Antti Karttunen, Oct 25 2014

Keywords

Links

Crossrefs

Bisection of A249152.
Cf. A002109, A007814, A143157, A069895 (first differences).

Programs

  • Mathematica
    Table[IntegerExponent[Hyperfactorial[2*n], 2], {n, 0, 55}] (* Amiram Eldar, Sep 10 2024 *)
  • Python
    from sympy import multiplicity
A249153_list, n = [0], 0
for i in range(2,20002,2):
    n += multiplicity(2,i)*i
    A249153_list.append(n) # Chai Wah Wu, Aug 21 2015

Formula

a(n) = A249152(2*n) = A007814(A002109(2*n)).
a(n) = 2*A143157(n).
a(n) ~ 2*n^2. - Amiram Eldar, Sep 10 2024

A246839 Number of trailing zeros in A002109(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 5, 5, 5, 5, 15, 15, 15, 15, 15, 30, 30, 30, 30, 30, 50, 50, 50, 50, 50, 100, 100, 100, 100, 100, 130, 130, 130, 130, 130, 165, 165, 165, 165, 165, 205, 205, 205, 205, 205, 250, 250, 250, 250, 250, 350, 350, 350, 350, 350, 405, 405, 405, 405
Offset: 0

Views

Author

Chai Wah Wu, Sep 04 2014

Keywords

Links

Crossrefs

Cf. A002109, A112765, A122840, A246817, A027868, A249152.

Programs

  • Mathematica
    (n=#;k=0;While[Mod[n,10]==0,n=n/10;k++];k)&/@Hyperfactorial@Range[0,60] (* Giorgos Kalogeropoulos, Sep 14 2021 *)
  • PARI
    a(n) = sum(i=1, n, i*valuation(i, 5)); \\ Michel Marcus, Sep 14 2021
  • Python
    def a(n):
  s = 1
  for k in range(n+1):
    s *= k**k
  i = 1
  while not s % 10**i:
    i += 1
  return i-1
n = 1
while n < 100:
  print(a(n),end=', ')
  n += 1 # Derek Orr, Sep 04 2014
  • Python
    from sympy import multiplicity
A246839, p5 = [0,0,0,0,0], 0
for n in range(5,10**3,5):
    p5 += multiplicity(5,n)*n
    A246839.extend([p5]*5)
# Chai Wah Wu, Sep 05 2014

Formula

From Michel Marcus, Sep 14 2021: (Start)
a(n) = A122840(A002109(n)), but also,
a(n) = A112765(A002109(n)), see explanation in A002109; so
a(n) = Sum_{i=1..n} i*v_5(i), where v_5(i) = A112765(i) is the exponent of the highest power of 5 dividing i. After a similar formula in A249152. (End)

A125760 a(n) = Product_{k=1..n} A002109(k).

Original entry on oeis.org

1, 1, 4, 432, 11943936, 1031956070400000, 4159895825138319360000000000, 13809882382682787973867537170432000000000000000, 769161257109634779902443718589603914508004789479014400000000000000000000, 16596916396875768196482032091931000424134701157007816971266990744831779993781534720000000000000000000000000
Offset: 0

Views

Author

N. J. A. Sloane, based on a suggestion from J. M. Bergot, Feb 06 2007

Keywords

Links

Crossrefs

Cf. A002109, A255269.

Programs

  • Maple
    seq(mul(mul(mul(k,j=1..k), k=1..m), m=1..n), n=0..9); # Zerinvary Lajos, Jun 01 2007
  • Mathematica
    Table[Product[Gamma[1 + k]^k/BarnesG[1 + k], {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Nov 19 2023 *)
Table[BarnesG[n + 2]^n/Product[BarnesG[k]^2, {k, 1, n + 1}], {n, 0, 10}] (* Vaclav Kotesovec, Nov 19 2023 *)

Formula

From Vaclav Kotesovec, Nov 19 2023: (Start)
a(n) = BarnesG(n+2)^n / Product_{k=1..n+1} BarnesG(k)^2.
a(n) ~ A^(n+1) * n^(n^3/6 + n^2/2 + 5*n/12 + 1/12) / exp(5*n^3/36 + n^2/4 + n/12 + zeta(3)/(4*Pi^2)), where A is the Glaisher-Kinkelin constant A074962. (End)

A246817 Possible number of trailing zeros in hyperfactorials (A002109).

Original entry on oeis.org

0, 5, 15, 30, 50, 100, 130, 165, 205, 250, 350, 405, 465, 530, 600, 750, 830, 915, 1005, 1100, 1300, 1405, 1515, 1630, 1750, 2125, 2255, 2390, 2530, 2675, 2975, 3130, 3290, 3455, 3625, 3975, 4155, 4340, 4530, 4725, 5125, 5330, 5540, 5755, 5975, 6425, 6655
Offset: 1

Views

Author

Chai Wah Wu, Sep 03 2014

Keywords

Comments

The number of trailing zeros in A002109 increases every 5 terms since the exponent of the factor 5 increases every 5 terms and the exponent of the factor 2 increases every 2 terms.

Links

Crossrefs

Cf. A002109, A191610, A246839.

Programs

  • Python
    from sympy import multiplicity
A246817, p5 = [0], 0
for n in range(5,5*10**3,5):
    p5 += multiplicity(5,n)*n
    A246817.append(p5) # Chai Wah Wu, Sep 05 2014

A219267 Logarithmic derivative of the hyperfactorials (A002109).

Original entry on oeis.org

1, 7, 313, 110143, 431860201, 24185951471887, 23238336572015738041, 445571476975584446962639039, 194201470505208674769594891331807753, 2157794122078406207016487628429579826176795887, 677208230450612019931822374477208301572175793625037599321
Offset: 1

Views

Author

Paul D. Hanna, Nov 16 2012

Keywords

Comments

Hyperfactorial A002109(n) = Product_{k=0..n} k^k.

Examples

			L.g.f.: L(x) = x + 7*x^2/2 + 313*x^3/3 + 110143*x^4/4 + 431860201*x^5/5 +...
where
exp(L(x)) = 1 + x + 4*x^2 + 108*x^3 + 27648*x^4 + 86400000*x^5 + 4031078400000*x^6 +...+ n^n*(n-1)^(n-1)*(n-2)^(n-2)*...*3^3*2^2*1^1*0^0**x^n +...

References

  • Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Crossrefs

Cf. A002109, A219266, A219268.

Programs

  • Mathematica
    nmax=15; Rest[CoefficientList[Series[Log[Sum[Product[j^j,{j,1,k}]*x^k,{k,0,nmax}]],{x,0,nmax}],x] * Range[0,nmax]] (* Vaclav Kotesovec, Jul 10 2015 *)
  • PARI
    {a(n)=n*polcoeff(log(sum(k=0,n+1,prod(j=0,k,j^j)*x^k)+x*O(x^n)),n)}
for(n=1,21,print1(a(n),", "))

Formula

a(n) ~ A * n^(n*(n+1)/2 + 13/12) / exp(n^2/4), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015

A264889 Partial sums of hyperfactorials (A002109).

Original entry on oeis.org

1, 2, 6, 114, 27762, 86427762, 4031164827762, 3319770429936027762, 55696441261496986915227762, 21577941278638297470665013744027762, 215779412250996503370318565758665013744027762, 61564384586850833363801728392684283449726665013744027762
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 27 2015

Keywords

Examples

			a(0) = 1;
a(1) = 1 + 1^1 = 2;
a(2) = 1 + 1^1 + 1^1*2^2 = 6;
a(3) = 1 + 1^1 + 1^1*2^2 + 1^1*2^2*3^3 = 114;
a(4) = 1 + 1^1 + 1^1*2^2 + 1^1*2^2*3^3 + 1^1*2^2*3^3*4^4 = 27762, etc.

Links

Crossrefs

Cf. A002109, A007489, A152690.

Programs

  • Mathematica
    Table[Sum[Hyperfactorial[k], {k, 0, n}], {n, 0, 11}]
Accumulate[Hyperfactorial[Range[0,15]]] (* Harvey P. Dale, Sep 22 2021 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=2, k, j^j)); \\ Altug Alkan, Nov 27 2015

Formula

a(n) = Sum_{k = 0..n} A002109(k).
a(n) = Sum_{k = 0..n} (k!)^k/Barnes G-Function(k + 1).
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