A081085 -id:A081085 - cp's OEIS frontend

A091401 Numbers n such that genus of group Gamma_0(n) is zero.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25

Offset: 1

N. J. A. Sloane, Mar 02 2004

Equivalently, numbers n such that genus of modular curve X_0(n) is zero.

G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.

Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A Harvey, Umbral moonshine and the Niemeier lattices, Research in the Mathematical Sciences, 2014, 1:3; See Eq. (22). - _N. J. A. Sloane_, Jun 19 2014
K. Harada, "Moonshine" of Finite Groups, European Math. Soc., 2010, p. 15.
Yang-Hui He, John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.
K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, vol. 102, American Mathematical Society, Providence, RI, 2004. See p. 110.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.

Cf. A001617, A001615, A000089, A000086, A001616, A091403.
The table below is a consequence of Theorem 7.3 in Maier's paper.
N        EntryID        K       alpha
1
      A127776        4096    1
      A276018        729     1
      A002894        256     1
      A276019        125     4
      A093388        72      1
      A276021        49      9
      A081085        32      1
      A006077        27      1
     A276020        20      2
     A276022        12      1
     A276177        13      36
     A276178        8       1
     A276179        6       1
     A276180        5       4

Flatten@ Position[#, 0] &@ Table[If[n < 1, 0, 1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors@ n}] - Count[(#^2 - # + 1)/n & /@ Range@ n, ?IntegerQ]/3 - Count[(#^2 + 1)/n & /@ Range@ n, ?IntegerQ]/4], {n, 120}] (* Michael De Vlieger, Dec 05 2016, after Michael Somos at A001617 *)

Numbers n such that A001617(n) = 0.

A060691 Expansion of AGM(1,1-8x) (where AGM denotes the arithmetic-geometric mean).

Original entry on oeis.org

1, -4, -4, -16, -84, -496, -3120, -20416, -137300, -942384, -6572336, -46432960, -331580272, -2389352256, -17351364160, -126851634432, -932823545428, -6895102385072, -51199649648048, -381738099675840, -2856639909232112

Offset: 0

Roland Bacher, Apr 20 2001

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A081085.

CoefficientList[Series[1/Hypergeometric2F1[1/2, 1/2, 1, 16*x*(1 - 4*x)], {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 13 2019 *)
CoefficientList[Series[Pi*(1 - 4*x)/(2*EllipticK[1/(1 - 1/(4*x))^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 13 2019 *)

a(n)=if(n<0,0,polcoeff(agm(1,1-8*x+x*O(x^n)),n))

G.f.: AGM(1, 1-8x).

a(n) ~ -Pi * 2^(3*n-1) / (n * log(n)^2) * (1 - (2*gamma + 4*log(2))/log(n) + (3*gamma^2 + 12*log(2)*gamma + 12*log(2)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

Edited by Michael Somos, Jul 19 2002

A053175 Catalan-Larcombe-French sequence.

Original entry on oeis.org

1, 8, 80, 896, 10816, 137728, 1823744, 24862720, 346498048, 4911669248, 70560071680, 1024576061440, 15008466534400, 221460239482880, 3287994183188480, 49074667327062016, 735814252604162048

Offset: 0

Peter J Larcombe, Nov 12 2001

These numbers were proposed as 'Catalan' numbers by an associate of Catalan. They appear as coefficients in the series expansion of an elliptic integral of the first kind. Defining f(x; c) = 1 /(1 - c^2*sin^2(x))^(1/2), consider the function I(c) obtained by integrating f(x; c) with respect to x between 0 and Pi/2. I(c) is transformed and written as a power series in c (through an intermediate variable) which acts as a generating function for the sequence.

Conjecture: Let P(n) be the (n+1) X (n+1) Hankel-type determinant with (i,j)-entry equal to a(i+j) for all i,j = 0,...,n. Then P(n)/2^(n*(n+3)) is a positive odd integer. - Zhi-Wei Sun, Aug 14 2013

			G.f. = 1 + 8*x + 80*x^2 + 896*x^3 + 10816*x^4 + 137728*x^5 + 1823774*x^6 + ...

P. J. Larcombe, D. R. French and E. J. Fennessey, The asymptotic behavior of the Catalan-Larcombe-French sequence {1, 8, 80, 896, 10816, ...}, Utilitas Mathematica, 60 (2001), 67-77.
P. J. Larcombe, D. R. French and C. A. Woodham, A note on the asymptotic behavior of a prime factor decomposition of the general Catalan-Larcombe-French number, Congressus Numerantium, 156 (2002), 17-25.

T. D. Noe, Table of n, a(n) for n=0..200
E. Catalan, Sur les Nombres de Segner, Rend. Circ. Mat. Pal., 1 (1887), 190-201. [From _Peter Luschny_, Jun 26 2009]
Lane Clark, An asymptotic expansion for the Catalan-Larcombe-French sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.1.
A. F. Jarvis, P. J. Larcombe and D. R. French, Linear recurrences between two recent integer sequences, Congressus Numerantium, 169 (2004), 79-99.
A. F. Jarvis, P. J. Larcombe and D. R. French, Applications of the a.g.m. of Gauss: some new properties of the Catalan-Larcombe-French sequence, Congressus Numerantium, 161 (2003), 151-162.
A. F. Jarvis, P. J. Larcombe and D. R. French, Power series identities generated by two recent integer sequences, Bulletin ICA, 43 (2005), 85-95.
A. F. Jarvis, P. J. Larcombe and D. R. French, On Small Prime Divisibility of the Catalan-Larcombe-French sequence, Indian Journal of Mathematics, 47 (2005), 159-181.
A. F. Jarvis, P. J. Larcombe and D. R. French, A short proof of the 2-adic valuation of the Catalan-Larcombe-French number, Indian Journal of Mathematics, 48 (2006), 135-138.
F. Jarvis, H. A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J (22) (2010) 171.
Xiao-Juan Ji, Zhi-Hong Sun, Congruences for Catalan-Larcombe-French numbers, arXiv:1505.00668 [math.NT], 2015 and JIS vol 19 (2016) # 16.3.4
P. J. Larcombe, A new asymptotic relation between two recent integer sequences, Congressus Numerantium, 175 (2005), 111-116.
Peter J. Larcombe, Daniel R. French, On the “Other” Catalan Numbers: A Historical Formulation Re-Examined, Congressus Numerantium, 143 (2000), 33-64.
P. J. Larcombe and D. R. French, On the integrality of the Catalan-Larcombe-French sequence {1, 8, 80, 896, 10816, ...}, Congressus Numerantium, 148 (2001), 65-91.
P. J. Larcombe and D. R. French, A new generating function for the Catalan-Larcombe-French sequence: proof of a result by Jovovic, Congressus Numerantium, 166 (2004), 161-172.
Guo-Shuai Mao, Proof of two supercongruences conjectured by Z.-W.Sun involving Catalan-Larcombe-French numbers, arXiv:1511.06222 [math.NT], 2015.
Brian Yi Sun, Baoyindureng Wu, Two-log-convexity of the Catalan-Larcombe-French sequence, arXiv:1602.04909 [math.CO], 2016. Also Journal of Inequalities and Applications, 2015, 2015:404; DOI: 10.1186/s13660-015-0920-0.
Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.
N. M. Temme, Examples of 3_F_2-polynomials, Asymptotic Methods for Integrals, Chapter 13, pp. 167-179 (2014).
Yang Wen, On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers, American Journal of Mathematical and Computer Modelling, 2017; 2(4): 95-98.
E. X. W. Xia and O. X. M. Yao, A Criterion for the Log-Convexity of Combinatorial Sequences, The Electronic Journal of Combinatorics, 20 (2013), #P3.

Cf. A065409, A002894, A081085.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 8 else (8*(3*n^2 -3*n+1)*a(n-1)-128*(n-1)^2*a(n-2))/n^2 fi end; # Peter Luschny, Jun 26 2009

a[ n_] := SeriesCoefficient[ EllipticK[ (8 x /(1 - 8 x))^2] / ((1 - 8 x) Pi/2), {x, 0, n}]; (* Michael Somos, Aug 01 2011 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ 8 x] BesselI[ 0, 4 x]^2, {x, 0, n}]]; (* Michael Somos, Aug 01 2011 *)
Table[(-8)^n Sqrt[Pi] HypergeometricPFQRegularized[{1/2, -n, -n}, {1, 1/2 - n}, -1]/n!, {n, 0, 20}] (* Vladimir Reshetnikov, May 21 2016 *)

{a(n) = if( n<0, 0, polcoeff( 1 / agm( 1, 1 - 16*x + x * O(x^n)), n))}; /* Michael Somos, Feb 12 2003 */

{a(n) = if( n<0, 0, polcoeff( sum( k=0, n, binomial( 2*k ,k)^2 * (2*x - 16*x^2)^k, x * O(x^n)), n))}; /* Michael Somos, Mar 04 2003 */

G.f.: 1 / AGM(1, 1 - 16*x) = 2 * EllipticK(8*x / (1-8*x)) / ((1-8*x)*Pi), where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre. Cf. A081085, A089602. - Michael Somos, Mar 04 2003 and Vladeta Jovovic, Dec 30 2003
E.g.f.: exp(8*x)*BesselI(0, 4*x)^2. - Vladeta Jovovic, Aug 20 2003
a(n)*n^2 = a(n-1)*8*(3*n^2 - 3*n + 1) - a(n-2)*128*(n-1)^2. - Michael Somos, Apr 01 2003
Exponential convolution of A059304 with itself: Sum(2^n*binomial(2*n, n)*x^n/n!, n=0..infinity)^2 = (BesselI(0, 4*x)*exp(4*x))^2 = hypergeom([1/2], [1], 8*x)^2. - Vladeta Jovovic, Sep 09 2003
a(n) ~ 2^(4n+1)/(Pi*n). - Vaclav Kotesovec, Oct 09 2012
a(n) = 2^n*Sum_{k=0..n} C(n,k)*C(2*k,k)*C(2(n-k),n-k), where C(n,k)=n!/(k!*(n-k)!). This formula has been proved via the Zeilberger algorithm (both sides of the equality satisfy the same recurrence relation). a(n)/2^n also has another expression: Sum_{k=0..floor(n/2)} C(n,2*k)*C(2*k,k)^2*4^(n-2*k). - Zhi-Wei Sun, Mar 21 2013
a(n) = (-1)^n*Sum_{k=0..n}C(2*k,k)*C(2(n-k),n-k)*C(k,n-k)*(-4)^k. I have proved this new formula via the Zeilberger algorithm. - Zhi-Wei Sun, Nov 19 2014

A362676 a(n) = Sum_{k = 0..n} 4^(n-k)binomial(n,k)binomial(n-1,k)binomial(2k,k).

Original entry on oeis.org

1, 4, 32, 328, 3840, 48504, 641984, 8765712, 122370048, 1736921560, 24975268032, 362872728816, 5317470233088, 78479369810352, 1165299414952320, 17393306836535328, 260791399517110272, 3925811865435871896, 59305018671515758784

Offset: 0

Peter Bala, Jul 03 2023

The sequence of Franel numbers A000172 satisfies the identity A000172(n) = Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*binomial(n+2*k,2*k)*binomial(2*k,k). The present sequence comes from the following modification of the right-hand side of the identity: a(n) = Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*binomial(-n+k,k)* binomial(2*k,k), multipled by a factor (-1)^n to give positive terms.
The Franel numbers satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r. We conjecture that the present sequence satisfies the same supercongruences.
From Peter Bala, Jul 07 2023 (Start):
Compare with the Domb numbers A002895, which are defined by A002895(n) = Sum_{k = 0..n} binomial(n,k)^2 * binomial(2*n-2*k,n-k) * binomial(2*k,k).
The supercongruences A002895(n*p^r) == A002895(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r (see Osburn and Sahu).
We conjecture that the present sequence satisfies the same supercongruences. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..833
Robert Osburn and Brundaban Sahu, A supercongruence for generalized Domb numbers, arXiv:1201.6195v2 [math.NT], Functiones et Approximatio. Comment. Math, Vol. 48, No 1, March 2013, 29-36.

Cf. A000172, A363985 - A363990, A002895, A081085, A364111.

seq(add(4^(n-k)*binomial(n,k)*binomial(n-1,k)*binomial(2*k,k), k = 0..n), n = 0..20);
# alternative faster program for large n
seq(simplify(4^n*hypergeom([-n, 1 - n, 1/2], [1, 1], 1)), n = 0..20);
# alternative (Peter Bala Jul 07 2023)
seq(add(binomial(n+k-1,k) * binomial(2*n-2*k,n-k) * binomial(2*k,k), k = 0..n), n = 0..20);

Table[4^n * HypergeometricPFQ[{-n, 1 - n, 1/2}, {1, 1}, 1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 04 2023 *)

from sympy import hyper, hyperexpand, S
def A362676(n): return int(hyperexpand(hyper((-n, 1-n, S.Half), [1,1], 1))*(1<<(n<<1))) # Chai Wah Wu, Jul 10 2023

a(n) = 4^n * hypergeom ([-n, 1 - n, 1/2], [1, 1], 1).
From Vaclav Kotesovec, Jul 04 2023: (Start)
Recurrence: (n-1)*n^2*(3*n^2 - 9*n + 7)*a(n) = 4*(n-1)*(15*n^4 - 60*n^3 + 80*n^2 - 40*n + 8)*a(n-1) - 4*(n-2)*(4*n - 7)*(4*n - 5)*(3*n^2 - 3*n + 1)*a(n-2).
a(n) ~ 2^(4*n - 1/2) / (Pi*n). (End)
a(n) = Sum_{k = 0..n} (-1)^k * binomial(-n,k) * binomial(2*n-2*k,n-k) * binomial(2*k,k). Cf. A081085. Peter Bala, Jul 07 2023
a(n) = binomial(2*n,n)*hypergeom([-n, n, 1/2], [1, 1/2 - n], 1). - Peter Bala, Jul 07 2023

A203576 Exponential (or binomial) half-convolution of A000984 (central binomial) with itself.

Original entry on oeis.org

1, 2, 14, 56, 446, 2152, 18248, 97120, 848254, 4796552, 42454664, 250140640, 2226532712, 13516860320, 120553738144, 748819997056, 6679690686334, 42254745008840, 376638926040392, 2418457241945056, 21530200591563496, 139992790135717792, 1244418656720926624, 8178446389043428736

Offset: 0

Wolfdieter Lang, Jan 13 2012

In general the exponential (also known as binomial) half-convolution of a sequence {b(n), n>=0} with itself is defined by
bhat(n) := Sum_{k=0..floor(n/2)} binomial(n,k)*b(k)*b(n-k), n>=0.
The e.g.f. of the sequence {bhat(n)} is Bhat(x) = ((B(x))^2 + B2(x^2))/2, with the e.g.f. B(x) of {b(n), n>=0} and the e.g.f. B2(x) := Sum_((b(n)^2/n!)*x^n/n!, n>=0) of the scaled squares. The proof runs along the same line as the one given for the ordinary half-convolution in a comment on A201204. In fact, bhat(n)/n! is the ordinary half-convolution of the sequence {b(n)/n!, n>=0} with itself.
Here b(n) = A000984(n) = binomial(2*n,n), n>=0, B(x) = exp(2*x)*BesselI(0,2*x) (see the Abramowitz-Stegun reference and link under A008277 for BesselI, p. 375, eq. 9.6.10) and B2(x) = hypergeometric([1/2,1/2],[1,1,1],16*x).

			With cbi = {1, 2, 6, 20, 70, 252, ...}
a(4) = 1*70 + 4*2*20 + 6*6^2 = 446,
a(5) = 1*252 + 5*2*70 + 10*6*20 = 2152.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000984, A081085 (exponential convolution).

cbi[n_] := Binomial[2*n, n]; a[n_] := Sum[Binomial[n, k]*cbi[k]*cbi[n - k], {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 09 2013 *)

A203576(n)=sum(k=0,n\2,binomial(n,k)*A000984(k)*A000984(n-k))  \\ M. F. Hasler, Jan 13 2012

a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*cbi(k)*cbi(n-k) n>=0, with cbi(n)=A000984(n).
E.g.f.: (exp(4*x)*BesselI(0, 2*x)^2 + hypergeom([1/2,1/2], [1,1,1],(4*x)^2))/2. See comment above.
Recurrence: (n-1)^2 * n^3 * (3*n^5 - 40*n^4 + 200*n^3 - 476*n^2 + 544*n - 241)*a(n) = 4*(n-1)^3 * (9*n^7 - 126*n^6 + 699*n^5 - 1997*n^4 + 3165*n^3 - 2770*n^2 + 1239*n - 228)*a(n-1) + 32*(3*n^10 - 55*n^9 + 420*n^8 - 1786*n^7 + 4731*n^6 - 8232*n^5 + 9630*n^4 - 7580*n^3 + 3900*n^2 - 1194*n + 162)*a(n-2) - 256*(n-2)^3 * (9*n^7 - 126*n^6 + 699*n^5 - 1997*n^4 + 3165*n^3 - 2770*n^2 + 1239*n - 228)*a(n-3) + 2048*(n-3)^3 * (n-2)^2 * (3*n^5 - 25*n^4 + 70*n^3 - 86*n^2 + 47*n - 10)*a(n-4). - Vaclav Kotesovec, Feb 25 2014
a(n) ~ 8^n / (Pi*n) * (1 + (1+(-1)^n)/sqrt(2*Pi*n)). - Vaclav Kotesovec, Feb 25 2014

A328046 G.f.: 1/2 + 1/(1 + AGM(1, sqrt(1-16*x))).

Original entry on oeis.org

1, 1, 7, 68, 763, 9276, 118656, 1572024, 21368155, 296187164, 4169180104, 59420124472, 855590919392, 12425933510200, 181787367119112, 2676258927443328, 39615617922076635, 589234154312057436, 8801406013366190952, 131964659304934491576, 1985338775295068132520

Offset: 0

Vaclav Kotesovec, Oct 03 2019

nonn

AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

Vaclav Kotesovec, Table of n, a(n) for n = 0..830
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean

Cf. A188267.

Cf. A060691, A081085, A323385.

CoefficientList[Series[1/2 + 1/(1 + (Pi*Sqrt[1 - 16*x])/(2*EllipticK[1 - 1/(1 - 16*x)])), {x, 0, 25}], x]

a(n) ~ Pi * 16^n / (n * (log(n) + Pi)^2) * (1 - (2*gamma + 8*log(2)) / (log(n) + Pi) + (3*gamma^2 + 48*log(2)^2 + 24*gamma*log(2) - Pi^2/2) / (log(n) + Pi)^2), where gamma is the Euler-Mascheroni constant A001620.

A228156 Expansion of sqrt((1+4x)/AGM(1+4x,1-4*x)) where AGM denotes the arithmetic-geometric mean.

Original entry on oeis.org

1, 2, 0, 8, 2, 68, 32, 720, 464, 8480, 6656, 106368, 95912, 1390928, 1392512, 18734144, 20371650, 257955716, 300101760, 3613109008, 4448177412, 51302395528, 66289160512, 736588435360, 992578330048, 10674012880512, 14924667774976, 155890890782720, 225244659392784, 2291995151532576, 3410654921389824

Offset: 0

Joerg Arndt, Aug 14 2013

nonn

Convolution square is A092266.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A092266 (1+4*x)/AGM(1+4*x,1-4*x).

Cf. A081085 1/AGM(1,1-8*x), A053175 1/AGM(1,1-16*x), A090004 1/AGM(1,1-16*x)^(1/2), A089602 1/AGM(1,1-16*x)^(1/4).

CoefficientList[Series[Sqrt[2*(1 + 4*x)*EllipticK[1 - (1 + 4*x)^2/(1 - 4*x)^2] / (Pi*(1 - 4*x))], {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 27 2019 *)

Vec( 1/agm(1,(1-4*x)/(1+4*x)+O(x^66))^(1/2) ) \\ Joerg Arndt, Aug 14 2013

a(n) ~ 2^(2*n - 1/2) / (n*sqrt(Pi*log(n))) * (1 - (gamma + 3*log(2)) / (2*log(n)) + (3*gamma^2/8 + 9*gamma*log(2)/4 + 27*log(2)^2/8 - 1/16*Pi^2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

cp's OEIS Frontend

A091401 Numbers n such that genus of group Gamma_0(n) is zero.

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A060691 Expansion of AGM(1,1-8x) (where AGM denotes the arithmetic-geometric mean).

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A053175 Catalan-Larcombe-French sequence.

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A362676 a(n) = Sum_{k = 0..n} 4^(n-k)*binomial(n,k)*binomial(n-1,k)*binomial(2*k,k).

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A203576 Exponential (or binomial) half-convolution of A000984 (central binomial) with itself.

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A328046 G.f.: 1/2 + 1/(1 + AGM(1, sqrt(1-16*x))).

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A228156 Expansion of sqrt((1+4*x)/AGM(1+4*x,1-4*x)) where AGM denotes the arithmetic-geometric mean.

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A362676 a(n) = Sum_{k = 0..n} 4^(n-k)binomial(n,k)binomial(n-1,k)binomial(2k,k).

A228156 Expansion of sqrt((1+4x)/AGM(1+4x,1-4*x)) where AGM denotes the arithmetic-geometric mean.