A000172 -id:A000172 - cp's OEIS frontend

A216353 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^3*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

1, 8, 532, 62624, 10964914, 2399234384, 609215149096, 171739556144192, 52316948995446679, 16918106849112020088, 5736533516906891508780, 2021549577502367744673888, 735516733692051220039803750, 274907827442478316252748869104, 105138174536582510069969443280760

Offset: 0

Paul D. Hanna, Sep 04 2012

nonn

			G.f.: A(x) = 1 + 8*x + 532*x^2 + 62624*x^3 + 10964914*x^4 + 2399234384*x^5 +...
where
log(A(x)) = 2^3*x + 10^3*x^2/2 + 56^3*x^3/3 + 346^3*x^4/4 + 2252^3*x^5/5 + 15184^3*x^6/6 + 104960^3*x^7/7 +...+ A000172(n)^3*x^n/n +...

Cf. A166990, A216352, 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)^3*x^m*1^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

A216355 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, 2, 175, 1760658, 1583078442003, 109611485085305859618, 547114144500297420116784959134, 189879050147329004652707990280499398833960, 4482752989702739533106941067588051779825642693578987967, 7097288803262045586874332782527584396862908242415791224663533782367102

Offset: 0

Paul D. Hanna, Sep 04 2012

nonn

			G.f.: A(x) = 1 + 2*x + 175*x^2 + 1760658*x^3 + 1583078442003*x^4 +...
where
log(A(x)) = 2*x + 346*x^2/2 + 5280932*x^3/3 + 6332299624282*x^4/4 + 548057409594239814752*x^5/5 +...+ A000172(n^2)*x^n/n +...

Cf. A216356, A166990, A216352, A216353, A216354, A000172.

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

A225776 Determinant of the (n+1) X (n+1) matrix with (i,j)-entry equal to f(i+j) for all i,j = 0,...,n, where f(k) = A000172(k) is the k-th Franel number.

Original entry on oeis.org

1, 6, 180, 28296, 23762160, 103179627360, 2242514387116224, 244558402519846478976, 136585911664795732792710912, 392586698202941899973146848809472, 5721548125375080140228462836137111413760

Offset: 0

Zhi-Wei Sun, Aug 14 2013

nonn

Conjecture: a(n)/6^n is always a positive odd integer. Moreover, for any integers r > 1 and n >= 0, the number a(r,n)/2^n is a positive odd integer, where a(r,n) denotes the Hankel determinant |f(r,i+j)|{i,j=0,...,n} with f(r,k) = sum{j=0}^k C(k,j)^r.

On Aug 20 2013, Zhi-Wei Sun made the following conjecture: If p is a prime congruent to 1 mod 4 but p is not congruent to 1 mod 24, then p divides a((p-1)/2).

			a(0) = 1 since f(0+0) = 1.

Zhi-Wei Sun, Table of n, a(n) for n = 0..25

Cf. A000172.

f[n_]:=Sum[Binomial[n,k]^3,{k,0,n}]; a[n_]:=Det[Table[f[i+j],{i,0,n},{j,0,n}]]; Table[a[n],{n,0,10}]

A088220 Coefficient of x^n in g.f.^n is A000172(n).

Original entry on oeis.org

1, 2, 3, 4, 9, 24, 75, 252, 903, 3376, 13068, 51960, 211222, 874440, 3676335, 15660680, 67474980, 293617248, 1288876879, 5701688928, 25397905302, 113838544880, 513117505278, 2324638603980, 10580591966824, 48362627748240

Offset: 0

Michael Somos, Sep 24 2003

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..550

Cf. A242903.

a(n)=polcoeff(x/serreverse(x*exp(sum(m=1, n+1, sum(k=0, m, binomial(m, k)^3)*x^m/m +x^2*O(x^n)))),n)
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, May 25 2014

G.f.: x / Series_Reversion( x*exp( Sum_{n>=1} A000172(n)*x^n/n ) ), where A000172(n) is the n-th Franel number. - Paul D. Hanna, May 25 2014

A199813 G.f.: exp( Sum_{n>=1} A000984(n)A000172(n) x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).

Original entry on oeis.org

1, 4, 38, 504, 8249, 154036, 3149326, 68741880, 1576163328, 37548785408, 922252542128, 23222906277952, 596981991939677, 15616173859832740, 414621835401615110, 11150969618415168280, 303278916800906999191, 8330190277527648516572, 230814933905555392525290

Offset: 0

Paul D. Hanna, Nov 10 2011

nonn

Sum_{k=0..n} C(n,k)^2 = A000984(n) defines central binomial coefficients.

Sum_{k=0..n} C(n,k)^3 = A000172(n) defines Franel numbers.

			G.f.: A(x) = 1 + 4*x + 38*x^2 + 504*x^3 + 8249*x^4 + 154036*x^5 +...
where
log(A(x)) = 2*2*x + 6*10*x^2/2 + 20*56*x^3/3 + 70*346*x^4/4 + 252*2252*x^5/5 + 924*15184*x^6/6 +...+ A000984(n)*A000172(n)*x^n/n +...

Cf. A199816, A181418, A000984, A000172.

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

A242903 G.f. A(x) satisfies: coefficient of x^n in A(x)^(2*n) equals A000172(n) = Sum_{k=0..n} C(n,k)^3, the n-th Franel number.

Original entry on oeis.org

1, 1, 1, 1, 3, 8, 26, 89, 324, 1225, 4786, 19170, 78408, 326275, 1377772, 5891401, 25467509, 111144579, 489145720, 2168854885, 9681072845, 43473716527, 196286934526, 890640262188, 4059500301390, 18579693200838, 85360357637580, 393548515741979, 1820335724153452, 8445294476235727, 39291407672079211

Offset: 0

Paul D. Hanna, May 25 2014

nonn

			G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 8*x^5 + 26*x^6 + 89*x^7 + 324*x^8 +...
Form a table of coefficients in A(x)^(2*n) as follows:
[1,  0,   0,    0,    0,     0,      0,      0,       0,       0, ...];
[1,  2,   3,    4,    9,    24,     75,    252,     903,    3376, ...];
[1,  4,  10,   20,   43,   108,    316,   1020,    3537,   12908, ...];
[1,  6,  21,   56,  138,   354,   1002,   3120,   10485,   37318, ...];
[1,  8,  36,  120,  346,   960,   2756,   8448,   27723,   96440, ...];
[1, 10,  55,  220,  735,  2252,   6785,  21020,   68340,  233870, ...];
[1, 12,  78,  364, 1389,  4716,  15184,  48588,  159186,  541424, ...];
[1, 14, 105,  560, 2408,  9030,  31304, 104960,  351792, 1203244, ...];
[1, 16, 136,  816, 3908, 16096,  60184, 213152,  739162, 2570464, ...];
[1, 18, 171, 1140, 6021, 27072, 109047, 409500, 1480293, 5280932, ...]; ...
then the main diagonal forms the Franel numbers:
[1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932, ...].

Cf. A166990, A088220, A000172.

{a(n)=polcoeff(sqrt(x/serreverse(x*exp(sum(m=1, n+1, sum(k=0, m, binomial(m, k)^3)*x^m/m +x^2*O(x^n))))),n)}
for(n=0,30,print1(a(n),", "))

G.f.: sqrt( x / Series_Reversion( x*exp( Sum_{n>=1} A000172(n)*x^n/n ) ) ), where A000172(n) is the n-th Franel number.
[x^n] A(x)^(2*n+2) = (n+1)*A166990(n).
Convolution square-root of A088220.

A058854 a(n) = largest prime in the factorization of n-th Franel number (A000172).

Original entry on oeis.org

2, 5, 7, 173, 563, 73, 41, 369581, 1409, 109, 449, 176459, 44221, 12148537, 148381, 11399977, 5779337237, 151431487, 26013917, 57405011, 939783003793, 277157, 191141, 13515438731, 79702499, 236463558839, 1883371283883863, 313527009031, 138961158000728258971

Offset: 1

Felix Goldberg (felixg(AT)tx.technion.ac.il), Jan 30 2001

nonn

			a(4)=173 because the 4th Franel number is 346 = 2^1 * 173^1, in which 173 is the largest prime.

Sean A. Irvine, Table of n, a(n) for n = 1..150

Cf. A000172.

with(combinat): with(numtheory): A000172 := n->sum(binomial(n,k)^3, k=0..n): for n from 1 to 50 do printf(`%d,`, sort(ifactors(A000172(n))[2])[nops(ifactors(A000172(n))[2])][1]) od: # Corrected by Sean A. Irvine, Aug 31 2022
# second Maple program:
a:= n-> max(numtheory[factorset](add(binomial(n, k)^3, k=0..n))):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 31 2022

Do[ Print[ FactorInteger[ Sum[ Binomial[n, k]^3, {k, 0, n}]] [[ -1, 1]] ], {n, 1, 32} ]

More terms from James Sellers, Feb 01 2001

Data corrected and entry revised by Sean A. Irvine, Aug 31 2022

A199816 G.f.: exp( Sum_{n>=1} A000984(n)A000172(n)/4 x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).

Original entry on oeis.org

1, 1, 8, 101, 1639, 30665, 630225, 13836981, 319062453, 7640441894, 188534274850, 4767113222750, 122998902095908, 3228067183537455, 85960229675478804, 2317956019913480326, 63193008693741620771, 1739473925024629613227, 48292271242981605779173

Offset: 0

Paul D. Hanna, Nov 11 2011

nonn

Sum_{k=0..n} C(n,k)^2 = A000984(n) defines central binomial coefficients.
Sum_{k=0..n} C(n,k)^3 = A000172(n) defines Franel numbers.
Compare to the g.f. of the Catalan numbers (A000108): exp(Sum_{n>=1} A000984(n)/2*x^n/n) and to the g.f. of A166991: exp(Sum_{n>=1} A000172(n)/2*x^n/n).

			G.f.: A(x) = 1 + x + 8*x^2 + 101*x^3 + 1639*x^4 + 30665*x^5 +...
where
log(A(x)) = 1*1*x + 3*5*x^2/2 + 10*28*x^3/3 + 35*173*x^4/4 + 126*1126*x^5/5 + 462*7592*x^6/6 +...+ A000984(n)/2*A000172(n)/2*x^n/n +...

Cf. A199813, A166991, A000108, A181418, A000984, A000172.

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

Convolution 4th power yields A199813.

A216356 a(n) = A000172(n^2), where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

Original entry on oeis.org

1, 2, 346, 5280932, 6332299624282, 548057409594239814752, 3282684865686445066146128050420, 1329153351023643434414727317328867397924832, 35862023917618878200052422822926970148356592776600354650, 63875599229358329592315180101212796802405282289343043273094466311541144

Offset: 0

Paul D. Hanna, Sep 04 2012

nonn

			L.g.f.: L(x) = 2*x + 346*x^2/2 + 5280932*x^3/3 + 6332299624282*x^4/4 + 548057409594239814752*x^5/5 +...
where exp(L(x)) = 1 + 2*x + 175*x^2 + 1760658*x^3 + 1583078442003*x^4 + 109611485085305859618*x^5 +...+ A216355(n)*x^n +...

Cf. A216355, A166990, A216352, A216353, A216354, A000172.

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

Forms the logarithmic derivative of A216355 after ignoring initial term a(0).

A207318 a(n) = Sum_{k=0..n-1} (-1)^k*A000172(k).

Original entry on oeis.org

0, 1, -1, 9, -47, 299, -1953, 13231, -91729, 647433, -4633499, 33531761, -244884159, 1802040241, -13346305519, 99392117841, -743734839215, 5588564785067, -42148760792553, 318928716891883, -2420342154102853, 18416484881248743, -140466988872011009, 1073705008744247231, -8223501739695527745

Offset: 0

N. J. A. Sloane, Feb 16 2012

sign

Z.-W. Sun, Congruences for Franel numbers, arXiv preprint arXiv:1112.1034, 2011. See Eq. 1.5.

Flatten[{0,Table[Sum[(-1)^k*Sum[Binomial[k,j]^3,{j,0,k}],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Jan 31 2014 *)

Conjecture: (n-1)^2*a(n) +(2*n-3)*(3*n-5)*a(n-1) +(-15*n^2+53*n-48)*a(n-2) +8*(n-2)^2*a(n-3)=0. - R. J. Mathar, Nov 28 2013

a(n) ~ (-1)^(n+1) * sqrt(3) * 2^(3*n+1) / (27*Pi*n). - Vaclav Kotesovec, Jan 31 2014

cp's OEIS Frontend

A216353 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^3*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

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A216355 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.

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A225776 Determinant of the (n+1) X (n+1) matrix with (i,j)-entry equal to f(i+j) for all i,j = 0,...,n, where f(k) = A000172(k) is the k-th Franel number.

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Mathematica

A088220 Coefficient of x^n in g.f.^n is A000172(n).

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Formula

A199813 G.f.: exp( Sum_{n>=1} A000984(n)*A000172(n) * x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).

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A242903 G.f. A(x) satisfies: coefficient of x^n in A(x)^(2*n) equals A000172(n) = Sum_{k=0..n} C(n,k)^3, the n-th Franel number.

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A058854 a(n) = largest prime in the factorization of n-th Franel number (A000172).

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Maple

Mathematica

Extensions

A199816 G.f.: exp( Sum_{n>=1} A000984(n)*A000172(n)/4 * x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).

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Formula

A216356 a(n) = A000172(n^2), where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

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A199813 G.f.: exp( Sum_{n>=1} A000984(n)A000172(n) x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).

A199816 G.f.: exp( Sum_{n>=1} A000984(n)A000172(n)/4 x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).