A246923 -id:A246923 - cp's OEIS frontend

A084771 Coefficients of expansion of 1/sqrt(1 - 10x + 9x^2); also, a(n) is the central coefficient of (1 + 5x + 4x^2)^n.

1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525, 653808673, 5614995765, 48416454529, 418895174885, 3634723102113, 31616937184725, 275621102802945, 2407331941640325, 21061836725455905, 184550106298084725

Offset: 0

Paul D. Hanna, Jun 10 2003

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps come in four colors and the H steps come in five colors. - N-E. Fahssi, Mar 30 2008
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and three kinds of steps (1,1). - Joerg Arndt, Jul 01 2011
Sums of squares of coefficients of (1+2*x)^n. - Joerg Arndt, Jul 06 2011
The Hankel transform of this sequence gives A103488. - Philippe Deléham, Dec 02 2007
Partial sums of A085363. - J. M. Bergot, Jun 12 2013
Diagonal of rational functions 1/(1 - x - y - 3*x*y), 1/(1 - x - y*z - 3*x*y*z). - Gheorghe Coserea, Jul 06 2018

			G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Tewodros Amdeberhan, In search of multiple expressions for a sequence
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
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.
Curtis Greene, Posets of shuffles, Journal of Combinatorial Theory, Series A 47.2 (1988): 191-206. See Eq. (30).
Christopher Huffaker, Nathan McCue, Cameron N. Miller, and Kayla S. Miller, The M&M Game: From Morsels to Modern Mathematics, arXiv:1508.06542 [math.HO], 2015.
Greg Morrow, Some probability distributions and integer sequences related to rook paths, Univ. Colorado Springs (2024). See p. 3.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
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. A001850, A059231, A059304, A246923 (a(n)^2).

List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*4^k)); # Muniru A Asiru, Jul 29 2018

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

seq(simplify(hypergeom([-n,1/2], [1], -8)),n=0..19); # Peter Luschny, Apr 26 2016

Table[n! SeriesCoefficient[E^(5 x) BesselI[0, 4 x], {x, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, May 10 2013 *)
Table[Hypergeometric2F1[-n, -n, 1, 4], {n,0,30}] (* Vladimir Reshetnikov, Nov 29 2013 *)
CoefficientList[Series[1/Sqrt[1-10x+9x^2],{x,0,30}],x] (* Harvey P. Dale, Mar 08 2016 *)
Table[3^n*LegendreP[n, 5/3], {n, 0, 40}] (* G. C. Greubel, May 30 2023 *)

{a(n) = if( n<0, -3 * 9^n * a(-1-n), sum(k=0,n, binomial(n, k)^2 * 4^k))}; /* Michael Somos, Oct 08 2003 */

{a(n) = if( n<0, -3 * 9^n * a(-1-n), polcoeff((1 + 5*x + 4*x^2)^n, n))}; /* Michael Somos, Oct 08 2003 */

/* as lattice paths: same as in A092566 but use */
steps=[[1,0], [0,1], [1,1], [1,1], [1,1]]; /* note the triple [1,1] */
/* Joerg Arndt, Jul 01 2011 */

a(n)={local(v=Vec((1+2*x)^n));sum(k=1,#v,v[k]^2);} /* Joerg Arndt, Jul 06 2011 */

a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])^2+imag(v[k])^2);} /* Joerg Arndt, Jul 06 2011 */

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

G.f.: 1 / sqrt(1 - 10*x + 9*x^2).
From Vladeta Jovovic, Aug 20 2003: (Start)
Binomial transform of A059304.
G.f.: Sum_{k >= 0} binomial(2*k,k)*(2*x)^k/(1-x)^(k+1).
E.g.f.: exp(5*x)*BesselI(0, 4*x). (End)
a(n) = Sum_{k = 0..n} Sum_{j = 0..n-k} C(n,j)*C(n-j,k)*C(2*n-2*j,n-j). - Paul Barry, May 19 2006
a(n) = Sum_{k = 0..n} 4^k*C(n,k)^2. - heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010
a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Sep 11 2012
D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2). - R. J. Mathar, Nov 26 2012
a(n) = hypergeom([-n, -n], [1], 4). - Vladimir Reshetnikov, Nov 29 2013
a(n) = hypergeom([-n, 1/2], [1], -8). - Peter Luschny, Apr 26 2016
From Michael Somos, Jun 01 2017: (Start)
a(n) = -3 * 9^n * a(-1-n) for all n in Z.
0 = a(n)*(+81*a(n+1) -135*a(n+2) +18*a(n+3)) +a(n+1)*(-45*a(n+1) +100*a(n+2) -15*a(n+3)) +a(n+2)*(-5*a(n+2) +a(n+3)) for all n in Z. (End)
From Peter Bala, Nov 13 2022: (Start)
1 + x*exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 5*x^2 + 29*x^3 + 185*x^4 + 1257*x^5 + ... is the g.f. of A059231.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all positive integers n and r and all primes p. (End)
a(n) = 3^n * LegendreP(n, 5/3). - G. C. Greubel, May 30 2023
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

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

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Oct 03 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 + 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)).

Seiichi Manyama, Table of n, a(n) for n = 0..481

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

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

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 *)

{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), ", "))

{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), ", "))

{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), ", "))

@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

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

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

Paul D. Hanna, Dec 10 2018

nonn

			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 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..481

Cf. A246467, A246906, A246923, A322246.

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 *)

/* 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), ", "))

/* 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), ", "))

/* 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), ", "))

/* 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), ", "))

/* 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), ", "))

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 + 5x + 16x^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

Paul D. Hanna, Dec 10 2018

nonn

			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 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A246467, A246906, A246923, A322248.

/* 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), ", "))

/* 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), ", "))

/* 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), ", "))

/* 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),", "))

/* 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), ", "))

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(1-64x, 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

cp's OEIS Frontend

A084771 Coefficients of expansion of 1/sqrt(1 - 10*x + 9*x^2); also, a(n) is the central coefficient of (1 + 5*x + 4*x^2)^n.

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

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

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

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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))).

A084771 Coefficients of expansion of 1/sqrt(1 - 10x + 9x^2); also, a(n) is the central coefficient of (1 + 5x + 4x^2)^n.

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

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

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

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

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