A248167 -id:A248167 - cp's OEIS frontend

A246467 G.f.: 1 / AGM(1-5x, sqrt((1-x)(1-25*x))).

1, 9, 121, 2025, 38025, 762129, 15912121, 341621289, 7484845225, 166549691025, 3751508008161, 85341068948529, 1957289174870121, 45199191579030225, 1049893021288265625, 24510327614556266025, 574726636455361317225, 13528549573868347823025, 319541915502909478890625

Offset: 0

Paul D. Hanna, Sep 06 2014

nonn

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.

			G.f.: A(x) = 1 + 9*x + 121*x^2 + 2025*x^3 + 38025*x^4 + 762129*x^5 +...
where the square-root of the terms yields A026375:
[1, 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, ...]
the g.f. of which is 1/sqrt((1-x)*(1-5*x)).

Seiichi Manyama, Table of n, a(n) for n = 0..717

Cf. A026375, A168597, A246876, A246906, A248167.

CoefficientList[Series[1/ArithmeticGeometricMean[1-5x,Sqrt[(1-x)(1-25x)]],{x,0,20}],x] (* Harvey P. Dale, Nov 01 2023 *)

{a(n)=polcoeff( 1 / agm(1-5*x, sqrt((1-x)*(1-25*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=sum(k=0,n,binomial(n,k)*binomial(2*k,k))^2}
for(n=0, 20, print1(a(n), ", "))

a(n) = A026375(n)^2 = [Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)]^2.
G.f.:  1 / AGM((1-x)*(1+5*x), (1+x)*(1-5*x)) = Sum_{n>=0} a(n)*x^(2*n).
a(n) ~ 5^(2*n+1) / (4*Pi*n). - Vaclav Kotesovec, Dec 10 2018

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

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Sep 07 2014

nonn

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.

			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)).

Seiichi Manyama, Table of n, a(n) for n = 0..593

Cf. A098409, A246467, A248167.

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 *)

{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), ", "))

{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), ", "))

{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), ", "))

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(1-9x, 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

Paul D. Hanna, Sep 07 2014

nonn

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.

			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)).

Seiichi Manyama, Table of n, a(n) for n = 0..525

Cf. A084771, A246467, A246906, A248167.

[9^n*Evaluate(LegendrePolynomial(n), 5/3)^2 : n in [0..40]]; // G. C. Greubel, May 30 2023

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 *)

{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), ", "))

{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), ", "))

{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), ", "))

[9^n*gen_legendre_P(n, 0, 5/3)^2 for n in range(41)] # G. C. Greubel, May 30 2023

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

A248168 Expansion of g.f. 1/sqrt((1-3x)(1-11*x)).

Original entry on oeis.org

1, 7, 57, 511, 4849, 47607, 477609, 4862319, 50026977, 518839783, 5414767897, 56795795679, 598213529809, 6322787125207, 67026654455433, 712352213507151, 7587639773475777, 80977812878889927, 865716569022673401, 9269461606674304959, 99387936492243451569, 1066975862517563301303

Offset: 0

Paul D. Hanna, Oct 03 2014

nonn

			G.f.: A(x) = 1 + 7*x + 57*x^2 + 511*x^3 + 4849*x^4 + 47607*x^5 +...
where A(x)^2 = 1/((1-3*x)*(1-11*x)):
A(x)^2 = 1 + 14*x + 163*x^2 + 1820*x^3 + 20101*x^4 + 221354*x^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..961
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 97.

Cf. A248167, A084771.

[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..40]]; // G. C. Greubel, May 31 2025

CoefficientList[Series[1/Sqrt[(1-3*x)*(1-11*x)], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 03 2014 *)

{a(n)=polcoeff( 1 / sqrt((1-3*x)*(1-11*x) +x*O(x^n)), n) }
for(n=0, 25, print1(a(n), ", "))

{a(n)=polcoeff( (1 + 7*x + 4*x^2 +x*O(x^n))^n, n) }
for(n=0, 25, print1(a(n), ", "))

{a(n)=sum(k=0,n, 3^(n-k)*2^k*binomial(n,k)*binomial(2*k,k))}
for(n=0, 25, print1(a(n), ", "))

@CachedFunction
def A248168(n):
     if (n<2): return 7^n
     else: return (7*(2*n-1)*A248168(n-1) - 33*(n-1)*A248168(n-2))//n
print([A248168(n) for n in range(41)]) # G. C. Greubel, May 31 2025

a(n) equals the central coefficient in (1 + 7*x + 4*x^2)^n, n>=0.
a(n) = Sum_{k=0..n} 3^(n-k) * 2^k * C(n,k) * C(2*k,k).
a(n) = Sum_{k=0..n} 11^(n-k) * (-2)^k * C(n,k) * C(2*k,k). - Paul D. Hanna, Apr 20 2019
a(n)^2 = A248167(n), which gives the coefficients in 1 / AGM(1-3*11*x, sqrt((1-3^2*x)*(1-11^2*x))).
Equals the binomial transform of 2^n*A026375(n).
Equals the second binomial transform of A084771.
Equals the third binomial transform of A059304(n) = 2^n*(2*n)!/(n!)^2.
a(n) ~ 11^(n+1/2)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 03 2014
D-finite with recurrence: n*a(n) +7*(-2*n+1)*a(n-1) +33*(n-1)*a(n-2)=0. [Belbachir]
a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 11^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

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

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 30 2019

nonn

See A246923.

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

Seiichi Manyama, Table of n, a(n) for n = 0..416

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).

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 *)

N=20; x='x+O('x^N); Vec(1/agm(1-64*x, sqrt((1-16*x)*(1-256*x))))

{a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))^2}

{a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))^2}

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(1-45x, sqrt((1-25x)(1-81x))).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 30 2019

nonn

See A246923.

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

Seiichi Manyama, Table of n, a(n) for n = 0..525

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).

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 *)

N=20; x='x+O('x^N); Vec(1/agm(1-45*x, sqrt((1-25*x)*(1-81*x))))

{a(n) = sum(k=0, n, 5^(n-k)*binomial(n, k)*binomial(2*k, k))^2}

{a(n) = sum(k=0, n, 9^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))^2}

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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A246467 G.f.: 1 / AGM(1-5*x, sqrt((1-x)*(1-25*x))).

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

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

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A248168 Expansion of g.f. 1/sqrt((1-3*x)*(1-11*x)).

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

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

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A246467 G.f.: 1 / AGM(1-5x, sqrt((1-x)(1-25*x))).

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

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

A248168 Expansion of g.f. 1/sqrt((1-3x)(1-11*x)).

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

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