A000172 -id:A000172 - cp's OEIS frontend

A260793 Primes p such that p does not divide any term of the Apéry-like sequence A000172 (also known as Type I primes).

3, 11, 17, 19, 43, 83, 89, 97, 113, 137, 139, 163, 193, 211, 233, 241, 283, 307, 313, 331, 347, 353, 379, 401, 409, 419, 433, 443, 491, 499, 523, 547, 569, 587, 601, 617, 619, 641, 643, 673, 811, 827, 859, 881, 929, 947, 953, 977, 1009, 1019, 1033, 1049, 1051

Offset: 1

N. J. A. Sloane, Aug 05 2015

nonn

See Schulte et al. (2014) for the precise definition of Type I primes.

Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
A. Schulte, S. VanSchalkwyk, A. Yang, On the divisibility and valuations of the Franel numbers, in MSRI-UP Research Reports, 2014.
A. Schulte, S. VanSchalkwyk, A. Yang, On the divisibility and valuations of the Franel numbers, Examples of Outstanding Student Posters, MAA.

Cf. A260791, A260792.

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see this sequence, A291275-A291284 and A133370 respectively.

maxPrime = 1051;
maxPi = PrimePi @ maxPrime;
okQ[p_] := AllTrue[Range[3 maxPi (* coeff 3 is empirical *)], GCD[HypergeometricPFQ[{-#, -#, -#}, {1, 1}, -1], p] == 1&];
Select[Prime[Range[maxPi]], okQ] (* Jean-François Alcover, Jan 13 2020 *)

Edited by N. J. A. Sloane, Aug 22 2017

A166990 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Original entry on oeis.org

1, 2, 7, 30, 147, 786, 4472, 26644, 164477, 1044258, 6782484, 44887236, 301782361, 2056250570, 14172792355, 98667874038, 692948001906, 4904403499992, 34951124337300, 250617829087656, 1807055528439771, 13095146839953030

Offset: 0

Paul D. Hanna, Nov 17 2009

nonn

Analogous to the square of the g.f. of Catalan numbers (A000108):

C(x)^2 = exp( Sum_{n>=1} A000984(n)*x^n/n ) where central binomial coefficient A000984(n) = Sum_{k=0..n} C(n,k)^2.

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 147*x^4 + 786*x^5 + 4472*x^6 +...
log(A(x)) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000172 (Franel numbers), A166991, A166992, A218117, A218119.

a[n_] := Sum[(Binomial[n, k])^3, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/n, {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* G. C. Greubel, May 30 2016 *)
nmax = 30; Clear[a]; franel = RecurrenceTable[{n^2*a[n] == (7*n^2 - 7*n + 2)*a[n-1] + 8*(n-1)^2*a[n-2], a[1] == 2, a[2] == 10}, a, {n, 1, nmax}]; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[franel[[k]]*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 27 2024 *)

{a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^3)*x^m/m)+x*O(x^n)),n)}

Self-convolution of A166991.

a(n) ~ c * 8^n / n^2, where c = 0.58462945... - Vaclav Kotesovec, Nov 27 2017, updated Oct 29 2024

A242169 Least prime divisor of Fr(n) which does not divide any Fr(k) with k < n, or 1 if such a primitive prime divisor of Fr(n) does not exist, where Fr(n) denotes the n-th Franel number given by A000172.

Original entry on oeis.org

2, 5, 7, 173, 563, 13, 41, 369581, 937, 61, 23, 29, 2141, 12148537, 31, 157, 59, 37, 506251, 151, 3019, 769, 47, 6730949, 79, 53, 3853, 661, 138961158000728258971, 1361, 421, 96920594213, 51378681049, 457, 71

Offset: 1

Zhi-Wei Sun, May 05 2014

nonn

Conjecture: a(n) > 1 for all n > 0.

			a(7) = 41 since Fr(7) = 2^9*5*41 with the prime factor 41 dividing none of Fr(1), ..., Fr(6) but 2 divides Fr(1) = 2 and 5 divides Fr(2) = 10.

Zhi-Wei Sun, Table of n, a(n) for n = 1..105

Cf. A000040, A000172, A242170, A242171, A242173.

Fr[n_]:=Sum[Binomial[n,k]^3,{k,0,n}]
f[n_]:=FactorInteger[Fr[n]]
p[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
Do[Do[Do[If[Mod[Fr[i],Part[p[n],k]]==0,Goto[aa]],{i,1,n-1}];Print[n," ",Part[p[n],k]];Goto[bb];Label[aa];Continue,{k,1,Length[p[n]]}];Print[n," ",1];Label[bb];Continue,{n,1,35}]

A232687 G.f. A(x) satisfies: the sum of the coefficients of x^k, k=0..n, in A(x)^n equals Sum_{k=0..n} C(n,k)^3 = A000172(n) (Franel numbers), for n>=0.

Original entry on oeis.org

1, 1, 3, 7, 20, 66, 244, 980, 4182, 18674, 86353, 410541, 1996214, 9888844, 49760925, 253767097, 1309154825, 6822023553, 35865392690, 190038440422, 1014015337209, 5444707218851, 29401289997403, 159584901816255, 870267544114291, 4766246752344215, 26206635040151511

Offset: 0

Paul D. Hanna, Dec 05 2013

nonn

Compare to: Sum_{k=0..n} [x^k] 1/(1-x)^n = Sum_{k=0..n} C(n,k)^2 = (2*n)!/n!^2.

a(n+1)/a(n) tends to 6.0295... - Vaclav Kotesovec, Jan 22 2014

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 20*x^4 + 66*x^5 + 244*x^6 + 980*x^7 +...
ILLUSTRATION OF INITIAL TERMS.
If we form an array of coefficients of x^k in A(x)^n, n>=0, like so:
A^0: [1],0,  0,   0,    0,    0,     0,      0,      0, ...;
A^1: [1, 1], 3,   7,   20,   66,   244,    980,   4182, ...;
A^2: [1, 2,  7], 20,   63,  214,   789,   3124,  13112, ...;
A^3: [1, 3, 12,  40], 138,  492,  1848,   7326,  30531, ...;
A^4: [1, 4, 18,  68,  255], 960,  3716,  14920,  62295, ...;
A^5: [1, 5, 25, 105,  425, 1691], 6785,  27805, 117165, ...;
A^6: [1, 6, 33, 152,  660, 2772, 11560], 48588, 207774, ...;
A^7: [1, 7, 42, 210,  973, 4305, 18676,  80746],351792, ...;
A^8: [1, 8, 52, 280, 1378, 6408, 28916, 128808, 573311], ...; ...
then the sum of the coefficients of x^k, k=0..n, in A(x)^n (shown above in brackets) equals Sum_{k=0..n} C(n,k)^3 = A000172(n):
A000172(0) = 1 = 1;
A000172(1) = 1 + 1 = 2;
A000172(2) = 1 + 2 +  7 = 10;
A000172(3) = 1 + 3 + 12 +  40 = 56;
A000172(4) = 1 + 4 + 18 +  68 + 255 = 346;
A000172(5) = 1 + 5 + 25 + 105 + 425 + 1691 = 2252;
A000172(6) = 1 + 6 + 33 + 152 + 660 + 2772 + 11560 = 15184; ...

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..460 (first 200 terms from Paul D. Hanna)

Cf. A232606, A232683, A232690, A000172.

Franel[n_] := Sum[Binomial[n, k]^3, {k, 0, n}];
a[0] = 1; a[n_] := Module[{B, G}, B = Sum[Franel[k]*x^k, {k, 0, n+1}] + x^3*O[x]^n; G = 1+x*O[x]^n; For[i=1, i <= n, i++, G = 1+Integrate[(B-1)* (G/x)-B*G^2, x]]; SeriesCoefficient[x/InverseSeries[x*G, x], {x, 0, n}]];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jan 15 2018, translated from 2nd PARI program *)

/* By Definition (slow): */
{Franel(n)=sum(k=0,n,binomial(n,k)^3)}
{a(n)=if(n==0, 1, (Franel(n) - sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j)^n + x*O(x^k), k)))/n)}
for(n=0, 20, print1(a(n), ", "))

/* Faster, using series reversion: */
{Franel(n)=sum(k=0,n,binomial(n,k)^3)}
{a(n)=local(B=sum(k=0, n+1, Franel(k)*x^k)+x^3*O(x^n), G=1+x*O(x^n));
for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); polcoeff(x/serreverse(x*G), n)}
for(n=0, 30, print1(a(n), ", "))

Given g.f. A(x), Sum_{k=0..n} [x^k] A(x)^n = Sum_{k=0..n} C(n,k)^3 = A000172(n).

Given g.f. A(x), let G(x) = A(x*G(x)) then (G(x) + x*G'(x)) / (G(x) - x*G(x)^2) = Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-2*x)^(3*n+1) = Sum_{n>=0} A000172(n)*x^n.

A166991 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)x^n/(2n) ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Original entry on oeis.org

1, 1, 3, 12, 57, 300, 1693, 10045, 61890, 392688, 2550843, 16891566, 113660475, 775223595, 5349057132, 37280705406, 262119009927, 1857241951359, 13250054817027, 95110710932424, 686490953423700, 4979704242810870

Offset: 0

Paul D. Hanna, Nov 17 2009

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1693*x^6 +...
log(A(x)^2) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000172 (Franel numbers), A166990, A166993, A218118, A218120.

a[n_] := Sum[(Binomial[n, k])^3, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/(2*n), {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* G. C. Greubel, May 30 2016 *)

{a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^3)/2*x^m/m)+x*O(x^n)),n)}

Self-convolution yields A166990.

a(n) ~ c * 8^n / n^2, where c = 0.231776... - Vaclav Kotesovec, Nov 27 2017

A248139 Least positive integer m such that m + n divides f(m) + f(n), where f(.) is given by A000172.

Original entry on oeis.org

1, 1, 25, 6, 14, 4, 13, 49, 19, 10, 2, 56, 2, 5, 6, 5, 27, 61, 9, 33, 23, 53, 21, 15, 3, 24, 11, 58, 39, 118, 3, 1598, 20, 40, 4, 2, 58, 26, 29, 17, 47, 34, 4, 31, 43, 163, 41, 25, 8, 26, 67, 40, 21, 214, 535, 12, 7, 22, 164, 74

Offset: 1

Zhi-Wei Sun, Oct 02 2014

nonn

Conjecture: a(n) exists for any n > 0.

			a(5) = 14 since 5 + 14 = 19 divides f(5) + f(14) = 2252 + 112738423360 = 112738425612 = 19*5933601348.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000172, A247824, A247937, A247940, A248124, A248125, A248133, A248136, A248137, A248142.

f[n_]:=Sum[Binomial[n,k]^3,{k,0,n}]
Do[m=1; Label[aa]; If[Mod[f[m]+f[n], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]

A216354 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^n*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Original entry on oeis.org

1, 2, 52, 58640, 3583098592, 11584364000042912, 2042518153012624794424576, 20047892010468651075834167466942080, 11138509206681372983092694151616405935206616064, 354938139483847646086359348765071470756626699510545192807936

Offset: 0

Paul D. Hanna, Sep 04 2012

nonn

			G.f.: A(x) = 1 + 2*x + 52*x^2 + 58640*x^3 + 3583098592*x^4 +...
where
log(A(x)) = 2*x + 10^2*x^2/2 + 56^3*x^3/3 + 346^4*x^4/4 + 2252^5*x^5/5 + 15184^6*x^6/6 + 104960^7*x^7/7 +...+ A000172(n)^n*x^n/n +...

Cf. A216355, A166990, A216352, A216353, A000172.

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3)^m*x^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

A052144 a(n) = A000172(n)^2.

Original entry on oeis.org

1, 4, 100, 3136, 119716, 5071504, 230553856, 11016601600, 546360462244, 27888242788624, 1456587070867600, 77515424509446400, 4189899499315360000, 229472379264509977600, 12709952101698593689600, 710863065714510068187136

Offset: 0

N. J. A. Sloane, Jan 23 2000

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 191.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A000172.

A052144[n_] := HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1]^2;
Array[A052144, 20, 0] (* Paolo Xausa, Jan 30 2024, after Jean-François Alcover in A000172 *)

P-recursive: P(n-1)*n^4*a(n) = P(n)*Q(n)*a(n-1) + 8*P(n-1)*Q(n)*a(n-2) - 512*P(n)*(n-2)^4*a(n-3), where P(n) = 7*n^2 - 7*n + 2 and Q(n) = 57*n^4 - 228*n^3 + 321*n^2 - 186*n + 40 with a(0) = 1, a(1) = 4 and a(2) = 100. - Peter Bala, Feb 01 2024

a(n) ~ 2^(6*n+2) / (3*Pi^2*n^2). - Vaclav Kotesovec, Feb 02 2024

A181418 a(n) = A000984(n)*A000172(n), which is the term-wise product of the Central binomial coefficients and Franel numbers, respectively.

Original entry on oeis.org

1, 4, 60, 1120, 24220, 567504, 14030016, 360222720, 9513014940, 256758913840, 7051260776560, 196403499277440, 5535202897806400, 157551884911456000, 4522682234563776000, 130783762623673221120, 3806221127760278029980

Offset: 0

Paul D. Hanna, Jan 28 2011

This sequence is s_6 in Cooper's paper. - Jason Kimberley, Nov 25 2012

Diagonal of the rational function R(x,y,z,w)=1/(1-(w*x*y+w*z+x*y+x*z+y+z)). - Gheorghe Coserea, Jul 13 2016

			E.g.f.: A(x) = 1 + 4*x/2! + 60*x^2/(2!*4!) + 1120*x^3/(3!*6!) + 24220*x^4/(4!*8!) + 567504*x^5/(5!*10!) +....
where A(x)^(1/2) = 1 + x + x^2/2!^3 + x^3/3!^3 + x^4/4!^3 +x^5/5!^3 +...

Jason Kimberley, Table of n, a(n) for n = 0..226
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.

Cf. A000984, A000172, A199813.

Related to diagonal of rational functions: A268545-A268555.

Table[Binomial[2n,n]*Sum[Binomial[n,k]^3,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Mar 06 2014 *)

{a(n)=binomial(2*n,n)*sum(k=0,n,binomial(n,k)^3)}

{a(n)=(2*n)!*n!*polcoeff(sum(m=0, n, x^m/m!^3+x*O(x^n))^2, n)}

a(n) = C(2n,n) * Sum_{k=0..n} C(n,k)^3.
E.g.f.: Sum_{n>=0} a(n)*x^n/(n!*(2*n)!) = ( Sum_{n>=0} x^n/n!^3 )^2.
From Jason Kimberley, Nov 26 2012: (Start)
1/Pi
  = (2/25)*Sum_{n>=0} a(n)*(9n+2)/50^n. [Cooper, equation (5)]
  = (2/25)*Sum_{n>=0} a(n)*A017185(n)/A165800(n). (End)
G.f.: 4*hypergeom([1/6, 1/3],[1],(27/2)*(1+(1-32*x)^(1/2))*(1-(1-32*x)^(1/2))^2/(3+(1-32*x)^(1/2))^3)^2/(3+(1-32*x)^(1/2)). - Mark van Hoeij, May 07 2013
Recurrence: n^3*a(n) = 2*(2*n-1)*(7*n^2 - 7*n + 2)*a(n-1) + 32*(n-1)*(2*n-3)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Mar 06 2014
a(n) ~ 2^(5*n+1) / (sqrt(3) * (Pi*n)^(3/2)). - Vaclav Kotesovec, Mar 06 2014
0 = (-x^2+28*x^3+128*x^4)*y''' + (-3*x+126*x^2+768*x^3)*y'' + (-1+92*x+864*x^2)*y' + (4+96*x)*y, where y is g.f. - Gheorghe Coserea, Jul 13 2016

A216352 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^2*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Original entry on oeis.org

1, 4, 58, 1256, 35771, 1200188, 45016678, 1827941560, 78753548245, 3551810922324, 166120394053698, 8002733850225288, 395089619067741926, 19911864121386482264, 1021345223473335336668, 53190166903606336969840, 2807000233813092463820488, 149869216802426305919295328

Offset: 0

Paul D. Hanna, Sep 04 2012

nonn

			G.f.: A(x) = 1 + 4*x + 58*x^2 + 1256*x^3 + 35771*x^4 + 1200188*x^5 +...
such that
log(A(x)) = 4*x + 100*x^2/2 + 3136*x^3/3 + 119716*x^4/4 + 5071504*x^5/5 +...+ A000172(n)^2*x^n/n +...

Cf. A166990, A216353, A216354, A216355, A052144, A000172.

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3)^2*x^m*1^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

cp's OEIS Frontend

A260793 Primes p such that p does not divide any term of the Apéry-like sequence A000172 (also known as Type I primes).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A166990 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A242169 Least prime divisor of Fr(n) which does not divide any Fr(k) with k < n, or 1 if such a primitive prime divisor of Fr(n) does not exist, where Fr(n) denotes the n-th Franel number given by A000172.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A232687 G.f. A(x) satisfies: the sum of the coefficients of x^k, k=0..n, in A(x)^n equals Sum_{k=0..n} C(n,k)^3 = A000172(n) (Franel numbers), for n>=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A166991 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/(2*n) ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A248139 Least positive integer m such that m + n divides f(m) + f(n), where f(.) is given by A000172.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A216354 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^n*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A052144 a(n) = A000172(n)^2.

Views

Author

Keywords

References

A166991 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)x^n/(2n) ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.