A201555 -id:A201555 - cp's OEIS frontend

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

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

Offset: 1

Ralf Steiner, Apr 18 2017

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

Indranil Ghosh, Table of n, a(n) for n = 1..40

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)).

[Numerator( n*(n^2+1)*Catalan(n^2)/2^(2*n^2-1) ): n in [1..21]]; // G. C. Greubel, Dec 11 2021

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 *)

A285388(n) = numerator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)); \\ Antti Karttunen, Apr 27 2017

a(n) = m=n*binomial(2*n^2, n^2);m>>valuation(m,2) \\ David A. Corneth, Apr 27 2017

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

[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

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

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

A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 15, 1, 1, 4, 70, 220, 1, 1, 5, 210, 5005, 4845, 1, 1, 6, 495, 48620, 735471, 142506, 1, 1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1, 1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1, 1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1

Offset: 0

Paul D. Hanna, Sep 04 2013

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

			The triangle of coefficients C(n*k, k^2), n>=k, k=0..n, begins:
1;
1, 1;
1, 2, 1;
1, 3, 15, 1;
1, 4, 70, 220, 1;
1, 5, 210, 5005, 4845, 1;
1, 6, 495, 48620, 735471, 142506, 1;
1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1;
1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1;
1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1; ...

Paul D. Hanna, Rows 0..30, as a flattened table of n, a(n) for n = 0..495

Cf. A228808 (row sums), A228833 (antidiagonal sums), A135860 (diagonal), A201555 (central terms).
Cf. A229052.
Cf. related triangles: A228904 (exp), A209330, A226234, A228836.

{T(n, k)=binomial(n*k, k^2)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A201556 G.f.: exp( Sum_{n>=1} C(2n^2,n^2) x^n/n ).

Original entry on oeis.org

1, 2, 37, 16278, 150303194, 25282422428664, 73752140616074524401, 3639659041645240391812731402, 2993893262520330875797362908273443346, 40656420461436928818704580402413441308206341488, 9054851465691640957562090101797213977192016103053025996396

Offset: 0

Paul D. Hanna, Dec 02 2011

nonn

Self-convolution of A213402.

Compare to the g.f. C(x) = 1 + x*C(x)^2 of the Catalan numbers (A000108): C(x)^2 = exp( Sum_{n>=1} binomial(2*n,n) * x^n/n ).

			G.f.: A(x) = 1 + 2*x + 37*x^2 + 16278*x^3 + 150303194*x^4 +...
where
log(A(x)) = 2*x + 70*x^2/2 + 48620*x^3/3 + 601080390*x^4/4 +...+ C(2*n^2,n^2)*x^n/n +...

G. C. Greubel, Table of n, a(n) for n = 0..40

Cf. A213402, A201555, A000984, A200002, A213409.

nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[2*k^2, k^2]*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b  (* Vaclav Kotesovec, Mar 06 2014 *)

{a(n)=polcoeff(exp(sum(m=1,n,binomial(2*m^2,m^2)*x^m/m)+x*O(x^n)),n)}

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,binomial(2*k^2,k^2)*a(n-k)))}

a(n) = (1/n) * Sum_{k=1..n} C(2*k^2,k^2) * a(n-k) for n>0 with a(0)=1.

a(n) ~ 4^(n^2) / (sqrt(Pi)*n^2). - Vaclav Kotesovec, Mar 06 2014

A285406 Base-2 logarithm of denominator of Sum_{k=0..n^2-1} (-1)^ksqrt(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

Ralf Steiner, Apr 18 2017

nonn

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

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

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 *)

a(n) = logint(denominator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)), 2); \\ Indranil Ghosh, Apr 27 2017

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

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

(define (A285406 n) (- (* 2 n n) (A007814 n) (A000120 (* n n)) 1)) ;; Antti Karttunen, Apr 28 2017

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

A285717 a(n) = A007814(n) + A159918(n) = A007814(n) + A000120(n^2).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 4, 3, 4, 5, 4, 4, 4, 4, 5, 3, 4, 5, 5, 6, 6, 3, 5, 5, 5, 6, 5, 5, 5, 5, 6, 3, 4, 5, 5, 6, 6, 7, 6, 5, 7, 7, 7, 8, 4, 4, 6, 5, 6, 5, 6, 8, 7, 7, 6, 6, 6, 7, 6, 6, 6, 6, 7, 3, 4, 5, 5, 6, 6, 7, 6, 6, 7, 9, 7, 7, 8, 5, 7, 6, 6, 8, 8, 7, 8, 7, 8, 9, 9, 5, 5, 6, 5, 5, 7, 5, 6, 6, 7, 9, 6, 7, 7, 6, 9, 8, 8, 8, 8, 4, 7, 7, 7, 8, 7, 9, 8, 8, 7

Offset: 1

Antti Karttunen, Apr 27 2017

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Cf. A000120, A007814, A159918, A201555, A285388, A285406.

Table[IntegerExponent[n, 2] + DigitCount[n^2, 2, 1], {n, 120}] (* Indranil Ghosh, Apr 27 2017 *)

a(n) = valuation(n, 2) + hammingweight(n^2); \\ Indranil Ghosh, Apr 27 2017

import math
def a(n): return int(math.log(n - (n & n - 1), 2)) + bin(n**2)[2:].count("1") # Indranil Ghosh, Apr 27 2017

(define (A285717 n) (+ (A007814 n) (A159918 n)))
(define (A285717 n) (A007814 (* n (A201555 n))))

a(n) = A007814(n) + A159918(n) = A007814(n) + A000120(n^2).

a(n) = A007814(n*A201555(n)).

A245245 a(n) = C(3*n^2, n^2).

Original entry on oeis.org

1, 3, 495, 4686825, 2254848913647, 52588547141148893628, 58152371703925106867047535565, 3012179439602547459232394950891834843500, 7255167425905233148164780983569428433097979870294255, 808718755397067598640202627155266231883064669446721506930287016061

Offset: 0

Paul D. Hanna, Jul 15 2014

nonn

a(n) = A005809(n^2).

Jens Kruse Andersen, Table of n, a(n) for n = 0..34

Cf. A245243, A201555, A005809, A213409 (exp).

Table[Binomial[3*n^2, n^2],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *)

{a(n)=binomial(3*n^2, n^2)}
for(n=0,15,print1(a(n),", "))

Logarithmic derivative of A213409 (ignoring initial term a(0) of this sequence).

a(n) ~ 3^(3*n^2+1/2) / (sqrt(Pi) * n * 2^(2*n^2+1)). - Vaclav Kotesovec, Jul 15 2014

A214441 Catalan numbers at square positions: a(n) = A000108(n^2).

Original entry on oeis.org

1, 1, 14, 4862, 35357670, 4861946401452, 11959798385860453492, 509552245179617138054608572, 368479169875816659479009042713546950, 4462290049988320482463241297506133183499654740, 896519947090131496687170070074100632420837521538745909320

Offset: 0

Paul D. Hanna, Jul 17 2012

nonn

Cf. A135758, A201555.

CatalanNumber[Range[0,10]^2] (* Harvey P. Dale, May 27 2013 *)

{a(n)=binomial(2*n^2,n^2)/(n^2+1)}
for(n=0,15,print1(a(n),", "))

a(n) = binomial(2*n^2, n^2) / (n^2 + 1).

A283048 a(n) = (6*n^2)!/((n^2)!^6).

Original entry on oeis.org

1, 720, 3246670537110000, 101097362223624462291180422369532000000, 11820969246826954556547676599863670334721199951925900837836206749590000

Offset: 0

Arkadiusz Wesolowski, Feb 27 2017

For n >= 1, a(n) is the number of possible combinations of an n X n X n Rubik's cube with randomly placed stickers.

Wikipedia, Rubik's Cube.

Cf. A201555.

[Factorial(6*n^2)/Factorial(n^2)^6: n in [0..4]];

Mathematica
```
Table[(6*n^2)!/(n^2)!^6, {n, 0, 4}]
```
PARI
```
a(n)=(6*n^2)!/(n^2)!^6;
```

a(n) = (6*n^2)!/((n^2)!^6).

cp's OEIS Frontend

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

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A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

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A201556 G.f.: exp( Sum_{n>=1} C(2*n^2,n^2) * x^n/n ).

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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).

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A285717 a(n) = A007814(n) + A159918(n) = A007814(n) + A000120(n^2).

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A245245 a(n) = C(3*n^2, n^2).

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A214441 Catalan numbers at square positions: a(n) = A000108(n^2).

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A201556 G.f.: exp( Sum_{n>=1} C(2n^2,n^2) x^n/n ).

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