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 1-8 of 8 results.

A246906 G.f.: 1 / AGM(1-21*x, sqrt((1-9*x)*(1-49*x))).

Original entry on oeis.org

1, 25, 729, 24025, 866761, 33350625, 1342856025, 55849505625, 2378365418025, 103099146750625, 4531090723144129, 201324497403240225, 9025111586043157801, 407581475160408424225, 18521763259935613598649, 846187436813348419025625, 38838031986984135802130025
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2014

Keywords

Comments

In general, the g.f. of the squares of coefficients in g.f. 1/sqrt((1-p*x)*(1-q*x)) is given by
1/AGM(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x))) = Sum_{n>=0} x^n*[Sum_{k=0..n} p^(n-k)*((q-p)/4)^k*C(n,k)*C(2*k,k)]^2,
and consists of integer coefficients when 4|(q-p).
Here AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 25*x + 729*x^2 + 24025*x^3 + 866761*x^4 +...
where the square-root of the terms yields A098409:
[1, 5, 27, 155, 931, 5775, 36645, 236325, 1542195, ...],
the g.f. of which is 1/sqrt((1-3*x)*(1-7*x)).

Links

Crossrefs

Cf. A098409, A246467, A248167.

Programs

  • Mathematica
    a[n_] := Sum[3^(n - k) * Binomial[n, k] * Binomial[2k, k], {k, 0, n}]^2; Array[a, 17, 0] (* Amiram Eldar, Dec 11 2018 *)
  • PARI
    {a(n,p=3,q=7)=polcoeff( 1 / agm(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x) +x*O(x^n))), n) }
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n,p=3,q=7)=polcoeff( 1 / sqrt((1-p*x)*(1-q*x) +x*O(x^n)), n)^2 }
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n,p=3,q=7)=sum(k=0,n,p^(n-k)*((q-p)/4)^k*binomial(n,k)*binomial(2*k,k))^2 }
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = A098409(n)^2 = [Sum_{k=0..n} 3^(n-k)*C(n,k)*C(2*k,k)]^2.
G.f.: 1 / AGM((1-3*x)*(1+7*x), (1+3*x)*(1-7*x)) = Sum_{n>=0} a(n)*x^(2*n).
From Vaclav Kotesovec, Sep 27 2019: (Start)
Recurrence: n^2*(2*n - 3)*a(n) = (2*n - 1)*(79*n^2 - 158*n + 54)*a(n-1) - 21*(2*n - 3)*(79*n^2 - 158*n + 54)*a(n-2) + 9261*(n-2)^2*(2*n - 1)*a(n-3).
a(n) ~ 7^(2*n+1) / (4*Pi*n). (End)

A246923 Expansion of g.f.: 1 / AGM(1-9*x, sqrt((1-x)*(1-81*x))).

Original entry on oeis.org

1, 25, 1089, 60025, 3690241, 241025625, 16359689025, 1140463805625, 81081830657025, 5852177325225625, 427465780890020929, 31528177440967935225, 2344153069158724611841, 175473167541934734763225, 13211212029033949825064769, 999630716942846408773325625
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2014

Keywords

Comments

In general, the g.f. of the squares of coefficients in g.f. 1/sqrt((1-p*x)*(1-q*x)) is given by
1/AGM(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x))) = Sum_{n>=0} x^n*( Sum_{k=0..n} p^(n-k)*((q-p)/4)^k*C(n,k)*C(2*k,k) )^2,
and consists of integer coefficients when 4|(q-p).
Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 25*x + 1089*x^2 + 60025*x^3 + 3690241*x^4 + 241025625*x^5 +...
where the square-root of each term yields A084771:
[1, 5, 33, 245, 1921, 15525, 127905, 1067925, ...],
the g.f. of which is 1/sqrt((1-x)*(1-9*x)).

Links

Crossrefs

Cf. A084771, A246467, A246906, A248167.

Programs

  • Magma
    [9^n*Evaluate(LegendrePolynomial(n), 5/3)^2 : n in [0..40]]; // G. C. Greubel, May 30 2023
  • Mathematica
    a[n_] := Sum[2^k * Binomial[n, k] * Binomial[2k, k], {k, 0, n}]^2; Array[a, 17, 0] (* Amiram Eldar, Dec 11 2018 *)
Table[9^n*LegendreP[n, 5/3]^2, {n, 0, 40}] (* G. C. Greubel, May 30 2023 *)
  • PARI
    {a(n,p=1,q=9)=polcoeff( 1 / agm(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x) +x*O(x^n))), n) }
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n,p=1,q=9)=polcoeff( 1 / sqrt((1-p*x)*(1-q*x) +x*O(x^n)), n)^2 }
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n,p=1,q=9)=sum(k=0,n,p^(n-k)*((q-p)/4)^k*binomial(n,k)*binomial(2*k,k))^2 }
for(n=0, 20, print1(a(n), ", "))
  • SageMath
    [9^n*gen_legendre_P(n, 0, 5/3)^2 for n in range(41)] # G. C. Greubel, May 30 2023

Formula

a(n) = A084771(n)^2 = ( Sum_{k=0..n} 2^k*C(n,k)*C(2*k,k) )^2.
G.f.: 1 / AGM((1-x)*(1+9*x), (1+x)*(1-9*x)) = Sum_{n>=0} a(n)*x^(2*n).
a(n) ~ 3^(4*n+2) / (8*Pi*n). - Vaclav Kotesovec, Sep 27 2019
a(n) = 9^n * ( LegendreP(n, 5/3) )^2. - G. C. Greubel, May 30 2023

A248167 Expansion of g.f.: 1 / AGM(1-33*x, sqrt((1-9*x)*(1-121*x))).

Original entry on oeis.org

1, 49, 3249, 261121, 23512801, 2266426449, 228110356881, 23642146057761, 2502698427758529, 269194720423487089, 29319711378381802609, 3225762406810715071041, 357859427246543331576481, 39977637030683399494792849, 4492572407488016429783217489, 507445676088537643607528136801
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2014

Keywords

Comments

In general, the g.f. of the squares of coefficients in g.f. 1/sqrt((1-p*x)*(1-q*x)) is given by 1/AGM(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x))) = Sum_{n>=0} x^n*( Sum_{k=0..n} p^(n-k)*((q-p)/4)^k*C(n,k)*C(2*k,k) )^2, and consists of integer coefficients when 4|(q-p). Here AGM(x, y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 49*x + 3249*x^2 + 261121*x^3 + 23512801*x^4 +...
where the square-root of the terms yields A248168:
[1, 7, 57, 511, 4849, 47607, 477609, 4862319, 50026977, ...],
the g.f. of which is 1/sqrt((1-3*x)*(1-11*x)).

Links

Crossrefs

Cf. A084771, A246467, A246906, A246923, A248168.

Programs

  • Magma
    m:=40;
A248168:= [n le 2 select 7^(n-1) else (7*(2*n-3)*Self(n-1) - 33*(n-2)*Self(n-2))/(n-1) : n in [1..m+2]];
A248167:= func< n | (A248168[n+1])^2 >;
[A248167(n): n in [0..m]]; // G. C. Greubel, May 31 2025
  • Mathematica
    a[n_] := Sum[3^(n - k) * 2^k * Binomial[n, k] * Binomial[2k, k], {k, 0, n} ]^2; Array[a, 17, 0] (* Amiram Eldar, Dec 11 2018 *)
  • PARI
    {a(n,p=3,q=11)=polcoeff( 1 / agm(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x) +x*O(x^n))), n) }
for(n=0, 20, print1(a(n,3,11), ", "))
  • PARI
    {a(n,p=3,q=11)=polcoeff( 1 / sqrt((1-p*x)*(1-q*x) +x*O(x^n)), n)^2 }
for(n=0, 20, print1(a(n,3,11), ", "))
  • PARI
    {a(n,p=3,q=11)=sum(k=0,n,p^(n-k)*((q-p)/4)^k*binomial(n,k)*binomial(2*k,k))^2 }
for(n=0, 20, print1(a(n,3,11), ", "))
  • SageMath
    @CachedFunction
def b(n): # b = A248168
     if (n<2): return 7^n
     else: return (7*(2*n-1)*b(n-1) - 33*(n-1)*b(n-2))//n
def A248167(n): return (b(n))^2
print([A248167(n) for n in range(41)]) # G. C. Greubel, May 31 2025

Formula

a(n) = A248168(n)^2 = ( Sum_{k=0..n} 3^(n-k)*2^k * C(n,k) * C(2*k,k) )^2.
G.f.: 1 / AGM((1-3*x)*(1+11*x), (1+3*x)*(1-11*x)) = Sum_{n>=0} a(n)*x^(2*n).
a(n) ~ 11^(2*n + 1) / (8*Pi*n). - Vaclav Kotesovec, Sep 27 2019

A246876 G.f.: 1 / AGM(1-12*x, sqrt((1-4*x)*(1-36*x))).

Original entry on oeis.org

1, 16, 324, 7744, 206116, 5875776, 175191696, 5386385664, 169300977444, 5410164352576, 175128910042384, 5727842622630144, 188931648862083856, 6276176070222305536, 209747841324097564224, 7046053064278540084224, 237764385841359952067364, 8054915184317632144620096
Offset: 0

Views

Author

Paul D. Hanna, Sep 06 2014

Keywords

Comments

In general, the g.f. of the squares of coefficients in g.f. 1/sqrt((1-p*x)*(1-q*x)) is given by
1/AGM(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x))) = Sum_{n>=0} x^n*[Sum_{k=0..n} p^(n-k)*((q-p)/4)^k*C(n,k)*C(2*k,k)]^2,
and consists of integer coefficients when 4|(q-p).
Here AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

Examples

			G.f.: A(x) = 1 + 16*x + 324*x^2 + 7744*x^3 + 206116*x^4 + 5875776*x^5 +...
where the square-root of the terms yields A081671:
[1, 4, 18, 88, 454, 2424, 13236, 73392, 411462, 2325976, ...]
the g.f. of which is 1/sqrt((1-2*x)*(1-6*x)).

Links

Crossrefs

Cf. A081671, A168597, A246467.

Programs

  • PARI
    {a(n)=polcoeff( 1 / agm(1-12*x, sqrt((1-4*x)*(1-36*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0,n,2^(n-k)*binomial(n,k)*binomial(2*k,k))^2}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = A081671(n)^2 = [Sum_{k=0..n} 2^(n-k) * C(n,k) * C(2*k,k)]^2.
G.f.: 1 / AGM((1-2*x)*(1+6*x), (1+2*x)*(1-6*x)) = Sum_{n>=0} a(n)*x^(2*n).
a(n) ~ 2^(2*n - 1) * 3^(2*n + 1) / (Pi*n). - Vaclav Kotesovec, Dec 10 2018

A322247 a(n) = A322246(n)^2, the square of the central coefficient in (1 + 5*x + 9*x^2)^n.

Original entry on oeis.org

1, 25, 1849, 156025, 14523721, 1426950625, 145317252025, 15178231605625, 1615509001626025, 174471431239950625, 19062335608125901729, 2102483602307417980225, 233721380163477368733481, 26154175972512598202392225, 2943361280244176889333396889, 332869229155486455718147125625, 37806108834415039621850996946025, 4310099976506176089944803738530625, 493021434686696395739629566004713025
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2018

Keywords

Examples

			G.f.: A(x) = 1 + 25*x + 1849*x^2 + 156025*x^3 + 14523721*x^4 + 1426950625*x^5 + 145317252025*x^6 + 15178231605625*x^7 + 1615509001626025*x^8 + ...
that is,
A(x) = 1 + 5^2*x + 43^2*x^2 + 395^2*x^3 + 3811^2*x^4 + 37775^2*x^5 + 381205^2*x^6 + 3895925^2*x^7 + 40193395^2*x^8 + 417697775^2*x^9 + ... + A322246(n)^2*x^n + ...

Links

Crossrefs

Cf. A246467, A246906, A246923, A322246.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(n-k) * 3^k * Binomial[n,k] * Binomial[2k,k], {k, 0, n}]^2; Array[a, 20, 0] (* Amiram Eldar, Dec 13 2018 *)
  • PARI
    /* a(n) = A322246(n)^2 - g.f. */
{a(n)=polcoeff(1/sqrt( (1 - x)*(1 + 11*x) +x*O(x^n)), n)^2}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* a(n) = A322246(n)^2 - g.f. */
{a(n) = polcoeff( (1 + 5*x + 9*x^2 +x*O(x^n))^n, n)^2}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* a(n) = A322246(n)^2 - binomial sum */
{a(n) = sum(k=0,n, (-1)^(n-k)*3^k*binomial(n,k)*binomial(2*k,k))^2}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* a(n) = A322246(n)^2 - binomial sum */
{a(n) = sum(k=0,n, 11^(n-k)*(-3)^k*binomial(n,k)*binomial(2*k,k))^2}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* Using AGM: */
{a(n)=polcoeff( 1 / agm(1 + 1*11*x, sqrt((1 - 1^2*x)*(1 - 11^2*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f.: 1 / AGM(1 + 11*x, sqrt((1 - x)*(1 - 11^2*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
G.f.: 1 / AGM((1-x)*(1-11*x), (1+x)*(1+11*x)) = Sum_{n>=0} a(n)*x^(2*n).
a(n) = A322248(n)^2, where A322248(n) = a(n) = Sum_{k=0..n} (-1)^(n-k) * 3^k * binomial(n,k)*binomial(2*k,k).
a(n) ~ 11^(2*n + 1) / (12*Pi*n). - Vaclav Kotesovec, Dec 13 2018

A322249 a(n) = A322248(n)^2, the square of the central coefficient in (1 + 5*x + 16*x^2)^n.

Original entry on oeis.org

1, 25, 3249, 366025, 48455521, 6646325625, 947789867025, 138422872355625, 20598606105401025, 3109408600719825625, 474780862425986767729, 73175222677868396505225, 11366022325041154078402081, 1777059915791491092441607225, 279404859303904515406536763089, 44144451113336819125597110875625, 7004264626817806908496520658161025, 1115512654966236899680358546064905625
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2018

Keywords

Examples

			G.f.: A(x) = 1 + 25*x + 3249*x^2 + 366025*x^3 + 48455521*x^4 + 6646325625*x^5 + 947789867025*x^6 + 138422872355625*x^7 + 20598606105401025*x^8 + ...
such that
A(x) = 1 + 5^2*x + 57^2*x^2 + 605^2*x^3 + 6961^2*x^4 + 81525^2*x^5 + 973545^2*x^6 + 11765325^2*x^7 + 143522145^2*x^8 + ... + A322248(n)^2*x^n + ...

Links

Crossrefs

Cf. A246467, A246906, A246923, A322248.

Programs

  • PARI
    /* a(n) = A322248(n)^2 - g.f. */
{a(n)=polcoeff(1/sqrt((1 + 3*x)*(1 - 13*x) +x*O(x^n)), n)^2}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* a(n) = A322248(n)^2 - g.f. */
{a(n) = polcoeff( (1 + 5*x + 16*x^2 +x*O(x^n))^n, n)^2}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* a(n) = A322248(n)^2 - binomial sum */
{a(n) = sum(k=0,n, (-3)^(n-k)*4^k*binomial(n,k)*binomial(2*k,k))^2}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* Using binomial formula: */
{a(n) = sum(k=0,n, 13^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))^2}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* Using AGM: */
{a(n)=polcoeff( 1 / agm(1 + 3*13*x, sqrt((1 - 3^2*x)*(1 - 13^2*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f.: 1 / AGM(1 + 3*13*x, sqrt((1 - 3^2*x)*(1 - 13^2*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
G.f.: 1 / AGM((1-3*x)*(1-13*x), (1+3*x)*(1+13*x)) = Sum_{n>=0} a(n)*x^(2*n).
a(n) = A322248(n)^2, where A322248(n) = a(n) = Sum_{k=0..n} (-3)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).
a(n) = A322248(n)^2, where A322248(n) = a(n) = Sum_{k=0..n} 13^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).
a(n) ~ 13^(2*n + 1) / (16*Pi*n). - Vaclav Kotesovec, Dec 10 2018

A307810 Expansion of 1/AGM(1-64*x, sqrt((1-16*x)*(1-256*x))).

Original entry on oeis.org

1, 100, 13924, 2371600, 453093796, 92598490000, 19745403216400, 4333667896360000, 971177275449892900, 221106619001508490000, 50967394891692703241104, 11866732390447357481358400, 2785834789480617203561744656, 658549235163074008904405646400
Offset: 0

Views

Author

Seiichi Manyama, Apr 30 2019

Keywords

Comments

See A246923.
Also the squares of coefficients in g.f. 1/sqrt((1-4*x)*(1-16*x)).

Links

Crossrefs

Cf. A307695.
(Sum_{k=0..n} c^(n-k)*e^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} d^(n-k)*(-e)^k*binomial(n,k)*binomial(2k,k))^2, where e = (d-c)/4: A002894 (c=0,d=4,e=1), A246467 (c=1,d=5,e=1), A246876 (c=2,d=6,e=1), A246906 (c=3,d=7,e=1), A307811 (c=5,d=9,e=1), A322240 (c=-3,d=5,e=2), A322243 (c=-1,d=7,e=2), A246923 (c=1,d=9,e=2), A248167 (c=3, d=11,e=2), A322247 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3), A322245 (c=-5,d=11,e=4), A322249 (c=-3,d=13,e=4).

Programs

  • Mathematica
    a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]^2; Array[a, 14, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
  • PARI
    N=20; x='x+O('x^N); Vec(1/agm(1-64*x, sqrt((1-16*x)*(1-256*x))))
  • PARI
    {a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))^2}
  • PARI
    {a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))^2}

Formula

a(n) = A307695(n)^2 = (Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k))^2.
a(n) ~ 2^(8*n+2) / (3*Pi*n). - Vaclav Kotesovec, Sep 27 2019

A307811 Expansion of 1/AGM(1-45*x, sqrt((1-25*x)*(1-81*x))).

Original entry on oeis.org

1, 49, 2601, 148225, 8970025, 570111129, 37678303881, 2567836387809, 179267329355625, 12754414737348025, 921185098227422161, 67340346156989933769, 4971327735657992896201, 369994703739586257235225, 27725052308247030792515625, 2089567204521186409129541025
Offset: 0

Views

Author

Seiichi Manyama, Apr 30 2019

Keywords

Comments

See A246923.
Also the squares of coefficients in g.f. 1/sqrt((1-5*x)*(1-9*x)).

Links

Crossrefs

Cf. A104454.
(Sum_{k=0..n} c^(n-k)*e^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} d^(n-k)*(-e)^k*binomial(n,k)*binomial(2k,k))^2, where e = (d-c)/4: A002894 (c=0,d=4,e=1), A246467 (c=1,d=5,e=1), A246876 (c=2,d=6,e=1), A246906 (c=3,d=7,e=1), this sequence (c=5,d=9,e=1), A322240 (c=-3,d=5,e=2), A322243 (c=-1,d=7,e=2), A246923 (c=1,d=9,e=2), A248167 (c=3, d=11,e=2), A322247 (c=-1,d=11,e=3), A307810 (c=4,d=16,e=3), A322245 (c=-5,d=11,e=4), A322249 (c=-3,d=13,e=4).

Programs

  • Mathematica
    a[n_] := Sum[5^(n-k) * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]^2; Array[a, 16, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
  • PARI
    N=20; x='x+O('x^N); Vec(1/agm(1-45*x, sqrt((1-25*x)*(1-81*x))))
  • PARI
    {a(n) = sum(k=0, n, 5^(n-k)*binomial(n, k)*binomial(2*k, k))^2}
  • PARI
    {a(n) = sum(k=0, n, 9^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))^2}

Formula

a(n) = A104454(n)^2 = (Sum_{k=0..n} 5^(n-k)*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} 9^(n-k)*(-1)^k*binomial(n,k)*binomial(2k,k))^2.
a(n) ~ 3^(4*n+2) / (4*Pi*n). - Vaclav Kotesovec, Sep 27 2019
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