A386621 -id:A386621 - cp's OEIS frontend

A098443 Expansion of 1/sqrt(1-8x-4x^2).

1, 4, 26, 184, 1366, 10424, 80996, 637424, 5064166, 40528984, 326251276, 2638751504, 21426682876, 174563719984, 1426219233416, 11681133293024, 95877105146246, 788433553532824, 6494463369141116, 53576199709855184

Offset: 0

Paul Barry, Sep 07 2004

Binomial transform of A098444. Second binomial transform of A084770. Third binomial transform of A098264.

			G.f. = 1 + 4*x + 26*x^2 + 184*x^3 + 1366*x^4 + 10424*x^5 + 80996*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Column k=2 of A386621.

Cf. A006139, A126869.

CoefficientList[Series[1/Sqrt[1 - 8*x - 4*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 15 2012, updated Mar 21 2024 *)

x='x+O('x^66); Vec(1/sqrt(1-8*x-4*x^2)) \\ Joerg Arndt, May 11 2013

E.g.f.: exp(4*x) * BesselI(0, 2*sqrt(5)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k) * binomial(2(n-k), n) * 2^(n-2k).
D-finite with recurrence: n*a(n) = 4*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ sqrt(50+20*sqrt(5))*(4+2*sqrt(5))^n/(10*sqrt(Pi*n)). Equivalently, a(n) ~ 2^(n-1/2) * phi^(3*n + 3/2) / (5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Oct 15 2012, updated Mar 21 2024
G.f.: 1/(1 - 2*x*(2+x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(2+x)/(k+1 - x*(2+x)*(2*k+2)*(4*k+3)/(2*x*(2+x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: Q(0), where Q(k) = 1 + 2*x*(x+2)*(4*k+1)/( 2*k+1 - x*(x+2)*(2*k+1)*(4*k+3)/(x*(x+2)*(4*k+3) + (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 16 2013
From Peter Bala, Mar 16 2024: (Start)
a(n) = (-2*i)^n * P(n, 2*i), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial.
Sum_{n >= 1} (-1)^(n+1)*4^n/(n*a(n-1)*a(n)) = 2*arctan(1/2) = 2*A073000. (End)
From Seiichi Manyama, Aug 29 2025: (Start)
a(n) = Sum_{k=0..n} (2-i)^k * (2+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.
a(n) = Sum_{k=0..floor(n/2)} 5^k * 4^(n-2*k) * binomial(n,2*k) * binomial(2*k,k).
a(n) = [x^n] (1+4*x+5*x^2)^n. (End)

A387430 a(n) = Sum_{k=0..n} (n-i)^k * (n+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.

Original entry on oeis.org

1, 2, 26, 576, 18886, 822800, 44758244, 2920443904, 222277449286, 19333107926208, 1891679562586252, 205658657276205056, 24594577004735218716, 3208651043895419972096, 453493188773477070618248, 69025100503218462336614400, 11256667883184684951198851654, 1958143582960886584057480612864

Offset: 0

Seiichi Manyama, Aug 29 2025

Sum_{k=0..n} (n-i)^k * (n+i)^(n-k) * binomial(n,k) = 2^n * n^n. - Vaclav Kotesovec, Aug 29 2025

Main diagonal of A386621.

Cf. A110129, A387459.

Join[{1}, Table[Sum[(n^2 + 1)^k * (2*n)^(n-2*k) * Binomial[n,2*k] * Binomial[2*k,k], {k,0,n/2}], {n,1,20}]] (* or *)
Table[(I + n)^n Hypergeometric2F1[-n, -n, 1, (-I + n)/(I + n)], {n, 0, 20}] (* Vaclav Kotesovec, Aug 29 2025 *)

a(n) = sum(k=0, n\2, (n^2+1)^k*(2*n)^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k));

a(n) = Sum_{k=0..floor(n/2)} n^(n-2*k) * binomial(2*(n-k),n-k) * binomial(n-k,k).
a(n) = Sum_{k=0..floor(n/2)} (n^2+1)^k * (2*n)^(n-2*k) * binomial(n,2*k) * binomial(2*k,k).
a(n) = [x^n] (1 + 2*n*x + (n^2+1)*x^2)^n.
a(n) ~ 2^(2*n) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Aug 29 2025

A387428 Expansion of 1/sqrt(1 - 12x - 4x^2).

Original entry on oeis.org

1, 6, 56, 576, 6216, 68976, 779456, 8921856, 103098976, 1200177216, 14053176576, 165337030656, 1952904166656, 23143984805376, 275062611081216, 3277130498482176, 39128126836409856, 468065027383059456, 5608576020851019776, 67305503084237193216, 808786974964809035776

Offset: 0

Seiichi Manyama, Aug 29 2025

Column k=3 of A386621.

a(n) = sum(k=0, n\2, 10^k*6^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k));

a(n) = Sum_{k=0..n} (3-i)^k * (3+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.
a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(2*(n-k),n-k) * binomial(n-k,k).
n*a(n) = 6*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 10^k * 6^(n-2*k) * binomial(n,2*k) * binomial(2*k,k).
a(n) = [x^n] (1+6*x+10*x^2)^n.
E.g.f.: exp(6*x) * BesselI(0, 2*sqrt(10)*x).
a(n) ~ 2^(n - 3/4) * (3 + sqrt(10))^(n + 1/2) / (5^(1/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 29 2025

A387466 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4x - (2k*x)^2).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 8, 20, 1, 2, 14, 32, 70, 1, 2, 24, 68, 136, 252, 1, 2, 38, 128, 406, 592, 924, 1, 2, 56, 212, 1096, 2332, 2624, 3432, 1, 2, 78, 320, 2566, 7632, 13964, 11776, 12870, 1, 2, 104, 452, 5320, 20092, 60864, 83848, 53344, 48620

Offset: 0

Seiichi Manyama, Aug 29 2025

			Square array begins:
    1,    1,     1,     1,      1,      1,       1, ...
    2,    2,     2,     2,      2,      2,       2, ...
    6,    8,    14,    24,     38,     56,      78, ...
   20,   32,    68,   128,    212,    320,     452, ...
   70,  136,   406,  1096,   2566,   5320,   10006, ...
  252,  592,  2332,  7632,  20092,  44752,   88092, ...
  924, 2624, 13964, 60864, 210524, 607424, 1523724, ...

Columns k=0..4 give A000984, A006139, A084770, A098455, A098456.
Main diagonal gives A387467.
Cf. A386621.

a(n, k) = sum(j=0, n\2, (k^2+1)^j*2^(n-2*j)*binomial(n, 2*j)*binomial(2*j, j));

A(n,k) = Sum_{j=0..n} (1-k*i)^j * (1+k*i)^(n-j) * binomial(n,j)^2, where i is the imaginary unit.
A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * binomial(2*(n-j),n-j) * binomial(n-j,j).
n*A(n,k) = 2*(2*n-1)*A(n-1,k) + 4*k^2*(n-1)*A(n-2,k) for n > 1.
A(n,k) = Sum_{j=0..floor(n/2)} (k^2+1)^j * 2^(n-2*j) * binomial(n,2*j) * binomial(2*j,j).
A(n,k) = [x^n] (1 + 2*x + (k^2+1)*x^2)^n.
E.g.f. of column k: exp(2*x) * BesselI(0, 2*sqrt(k^2+1)*x).

cp's OEIS Frontend

A098443 Expansion of 1/sqrt(1-8x-4x^2).

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A387430 a(n) = Sum_{k=0..n} (n-i)^k * (n+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.

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A387428 Expansion of 1/sqrt(1 - 12x - 4x^2).

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A387466 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4x - (2k*x)^2).

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cp's OEIS Frontend

A098443 Expansion of 1/sqrt(1-8*x-4*x^2).

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Mathematica

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Formula

A387430 a(n) = Sum_{k=0..n} (n-i)^k * (n+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.

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A387428 Expansion of 1/sqrt(1 - 12*x - 4*x^2).

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A387466 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4*x - (2*k*x)^2).

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A098443 Expansion of 1/sqrt(1-8x-4x^2).

A387428 Expansion of 1/sqrt(1 - 12x - 4x^2).

A387466 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4x - (2k*x)^2).