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.

A285388 a(n) = numerator of ((1/n) * Sum_{k=0..n^2-1} binomial(2k,k)/4^k).

Original entry on oeis.org

1, 35, 36465, 300540195, 79006629023595, 331884405207627584403, 22292910726608249789889125025, 11975573020964041433067793888190275875, 411646257111422564507234009694940786177843149765, 56592821660064550728377610673427602421565368547133335525825
Offset: 1

Views

Author

Ralf Steiner, Apr 18 2017

Keywords

Comments

Editorial comment: This sequence arose from Ralf Steiner's attempt to prove Legendre's conjecture that there is a prime between N^2 and (N+1)^2 for all N. - N. J. A. Sloane, May 01 2017

Links

Crossrefs

Cf. A000079, A000265, A056220, A060757, A201555, A285389 (denominators), A285406, A280655 (similar), A190732 (2/sqrt(Pi)), A285738 (greatest prime factor), A285717, A285730, A285786, A286264, A000290 (n^2), A056220 (2*n^2 -1), A286127 (sum a(n-1)/a(n)).

Programs

  • Magma
    [Numerator( n*(n^2+1)*Catalan(n^2)/2^(2*n^2-1) ): n in [1..21]]; // G. C. Greubel, Dec 11 2021
  • Mathematica
    Table[Numerator[Sum[Binomial[2k,k]/4^k,{k,0,n^2-1}]/n],{n,1,10}]
Numerator[Table[2^(1-2 n^2) n Binomial[2 n^2,n^2],{n,1,10}]] (* Ralf Steiner, Apr 22 2017 *)
  • PARI
    A285388(n) = numerator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)); \\ Antti Karttunen, Apr 27 2017
  • PARI
    a(n) = m=n*binomial(2*n^2, n^2);m>>valuation(m,2) \\ David A. Corneth, Apr 27 2017
  • Python
    from sympy import binomial, Integer
def a(n): return (Integer(2)**(1 - 2*n**2)*n*binomial(2*n**2, n**2)).numerator # Indranil Ghosh, Apr 27 2017
  • Sage
    [numerator( n*(n^2+1)*catalan_number(n^2)/2^(2*n^2-1) ) for n in (1..20)] # G. C. Greubel, Dec 11 2021

Formula

a(n) is numerator of n*binomial(2 n^2, n^2)/2^(2*n^2 - 1). - Ralf Steiner, Apr 26 2017
a(n) = numerator(n*A201555(n) / (A060757(n)/2)) = n*A201555(n) / 2^(A285717(n)) = A000265(n*A201555(n)). [Using Ralf Steiner's formula and A285717(n) <= A056220(n), cf. A285406.] - Antti Karttunen, Apr 27 2017
Limit_{i->oo} a(i)*A285389(i+1)/(a(i+1)*A285389(i)) = 1. - Ralf Steiner, May 03 2017

Extensions

Edited (including the removal of the author's claim that this leads to a proof of the Legendre conjecture) by N. J. A. Sloane, May 01 2017
Formula section edited by M. F. Hasler, May 02 2017
Edited by N. J. A. Sloane, May 10 2017

A201555 a(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.

Original entry on oeis.org

1, 2, 70, 48620, 601080390, 126410606437752, 442512540276836779204, 25477612258980856902730428600, 23951146041928082866135587776380551750, 365907784099042279561985786395502921046971688680, 90548514656103281165404177077484163874504589675413336841320
Offset: 0

Views

Author

Paul D. Hanna, Dec 02 2011

Keywords

Comments

Central coefficients of triangle A228832.

Examples

			L.g.f.: L(x) = 2*x + 70*x^2/2 + 48620*x^3/3 + 601080390*x^4/4 + ...
where exponentiation equals the g.f. of A201556:
exp(L(x)) = 1 + 2*x + 37*x^2 + 16278*x^3 + 150303194*x^4 + ... + A201556(n)*x^n + ...

Links

Crossrefs

Cf. A000984, A007814, A159918, A201556, A214441, A228832, A285388, A285717.

Programs

  • Mathematica
    Table[Binomial[2n^2,n^2],{n,0,10}] (* Harvey P. Dale, Dec 10 2011 *)
  • PARI
    a(n) = binomial(2*n^2,n^2)
  • Python
    from math import comb
def A201555(n): return comb((m:=n**2)<<1,m) # Chai Wah Wu, Jul 08 2022

Formula

L.g.f.: ignoring initial term, equals the logarithmic derivative of A201556.
a(n) = (2*n^2)! / (n^2)!^2.
a(n) = Sum_{k=0..n^2} binomial(n^2,k)^2.
For primes p >= 5: a(p) == 2 (mod p^3), Oblath, Corollary II; a(p) == binomial(2*p,p) (mod p^6) - see Mestrovic, Section 5, equation 31. - Peter Bala, Dec 28 2014
A007814(a(n)) = A159918(n). - Antti Karttunen, Apr 27 2017, based on Vladimir Shevelev's Jul 20 2009 formula in A000984.

A285406 Base-2 logarithm of denominator of Sum_{k=0..n^2-1} (-1)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k)*n).

Original entry on oeis.org

0, 5, 15, 28, 46, 68, 94, 123, 158, 195, 236, 283, 333, 387, 445, 506, 574, 643, 716, 794, 875, 961, 1054, 1146, 1244, 1346, 1451, 1562, 1676, 1794, 1916, 2041, 2174, 2307, 2444, 2586, 2731, 2881, 3034, 3193, 3356, 3520, 3690, 3864, 4041, 4227, 4413, 4601, 4796, 4993
Offset: 1

Views

Author

Ralf Steiner, Apr 18 2017

Keywords

Comments

Needed for studying of Wallis-kind products of central binomials.

Crossrefs

Cf. A000120, A000523, A000984, A007814, A011371, A056220, A201555, A285389, A281264, A285717.

Programs

  • Mathematica
    Log[2,Table[Denominator[Sum[Binomial[2k,k]/4^k,{k,0,n^2-1}]/n], {n,1,50}]]
Log[2,Denominator[Table[2^(1-2 n^2) n Binomial[2 n^2,n^2],{n,1,50}]]] (* Ralf Steiner, Apr 22 2017 *)
  • PARI
    a(n) = logint(denominator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)), 2); \\ Indranil Ghosh, Apr 27 2017
  • PARI
    val(n, p) = my(r=0); while(n, r+=n\=p);r
a(n) = 2*n^2-1 - valuation(n, 2) - val(2*n^2, 2) + 2*val(n^2, 2) \\ David A. Corneth, Apr 28 2017
  • Python
    from sympy import binomial, integer_log, Integer
def a(n): return integer_log((Integer(2)**(1 - 2*n**2)*n*binomial(2*n**2, n**2)).denominator, 2)[0] # Indranil Ghosh, Apr 27 2017
  • Scheme
    (define (A285406 n) (- (* 2 n n) (A007814 n) (A000120 (* n n)) 1)) ;; Antti Karttunen, Apr 28 2017

Formula

a(n) = A000523(A285389(n)).
a(n) = A056220(n) - A285717(n) = (2*(n^2)) - A007814(n) - A000120(n^2) - 1. - Antti Karttunen, Apr 28 2017, based on Vladimir Shevelev's Jul 20 2009 formula in A000984
Showing 1-3 of 3 results.

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

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