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
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)).
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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
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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 *)
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{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), ", "))
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{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), ", "))
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{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), ", "))
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@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
A322248
G.f.: 1/sqrt( (1 + 3*x)*(1 - 13*x) ).
Original entry on oeis.org
1, 5, 57, 605, 6961, 81525, 973545, 11765325, 143522145, 1763351525, 21789466777, 270509191485, 3371353189009, 42155188480085, 528587607974217, 6644129071092525, 83691484792766145, 1056178325362832325, 13351036742005533945, 169019946403985898525, 2142600388730167543281, 27193744661180635582005, 345520219114720175821737, 4394534009569783690837005, 55943630366450131877449761, 712778930909503993783945125
Offset: 0
G.f.: A(x) = 1 + 5*x + 57*x^2 + 605*x^3 + 6961*x^4 + 81525*x^5 + 973545*x^6 + 11765325*x^7 + 143522145*x^8 + 1763351525*x^9 + 21789466777*x^10 + ...
such that A(x)^2 = 1/(1 - 10*x - 39*x^2).
RELATED SERIES.
exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 5*x + 41*x^2 + 365*x^3 + 3537*x^4 + 35925*x^5 + 378105*x^6 + 4084925*x^7 + 45044129*x^8 + 504880805*x^9 + 5735247817*x^10 + ...
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CoefficientList[Series[1/Sqrt[((1+3x)(1-13x))],{x,0,30}],x] (* Harvey P. Dale, Jun 29 2021 *)
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/* Using generating function: */
{a(n) = polcoeff( 1/sqrt((1 + 3*x)*(1 - 13*x) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))
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/* Using binomial formula: */
{a(n) = sum(k=0,n, (-3)^(n-k)*4^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))
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/* Using binomial formula: */
{a(n) = sum(k=0,n, 13^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))
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/* a(n) as a central coefficient */
{a(n) = polcoeff( (1 + 5*x + 16*x^2 +x*O(x^n))^n, n)}
for(n=0,30,print1(a(n),", "))
A307695
Expansion of 1/(sqrt(1-4*x)*sqrt(1-16*x)).
Original entry on oeis.org
1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300, 225759595348, 3444812388280, 52781007848284, 811510465220920, 12513859077134008, 193460383702061200, 2997463389599395270, 46532910920993515900, 723626591914643806180, 11270311875128088314200
Offset: 0
Cf.
A000984 (c=0,d=4,e=1),
A026375 (c=1,d=5,e=1),
A081671 (c=2,d=6,e=1),
A098409 (c=3,d=7,e=1),
A098410 (c=4,d=8,e=1),
A104454 (c=5,d=9,e=1).
Cf.
A084605 (c=-3,d=5,e=2),
A098453 (c=-2,d=6,e=2),
A322242 (c=-1,d=7,e=2),
A084771 (c=1,d=9,e=2),
A248168 (c=3,d=11,e=2).
Cf.
A322246 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3).
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a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
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N=66; x='x+O('x^N); Vec(1/sqrt(1-20*x+64*x^2))
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{a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))}
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{a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))}
Showing 1-3 of 3 results.
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