A317651 -id:A317651 - cp's OEIS frontend

A362714 a(0) = 1 and a(n) = 2^(n-1)Product_{j=1..n} (4j - 3)^2 - Sum_{m=1..n-1} binomial(2n, 2m)a(m)a(n-m)/2 for n > 0.

1, 1, 47, 7395, 2453425, 1399055625, 1221037941375, 1513229875486875, 2526879997358510625, 5469272714829657020625, 14892997153152592003359375, 49826568404835717359311321875, 200913471834337931507493300140625, 960945974809003219596852282787265625, 5378917217051713436481068409370884609375

Offset: 0

Stefano Spezia, Apr 30 2023

nonn

Christian Krattenthaler and Thomas W. Müller, The congruence properties of Romik's sequence of Taylor coefficients of Jacobi's theta function theta_3, arXiv:2304.11471 [math.NT], 2023. See p. 6.

Cf. A317651, A362713, A362715.

a[0]=1; a[n_]:=2^(n-1)Product[(4j-3)^2,{j,n}]-Sum[Binomial[2n,2m]a[m]a[n-m],{m,n-1}]/2; Array[a,15,0]
nmax = 20; Table[(k-1)! * 2^((k-1)/2) * CoefficientList[Series[Sqrt[Hypergeometric2F1[1/4, 1/4, 1/2, 4*x^2]], {x, 0, 2*nmax+2}], x][[k]], {k, 1, 2*nmax+2, 2}] (* Vaclav Kotesovec, May 03 2023 *)

E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2^n*(2*n)!) = sqrt(2F1([1/4, 1/4], [1/2], 4*x^2)).

A362715 Triangle read by rows: T(n, k) = 2^(n-k)(2n)!/(2k)! [x^(2n)] U[x]^(2k), where U(x) = x2F1([3/4, 3/4], [3/2], 4x^2)/2F1([1/4, 1/4], [1/2], 4*x^2).

Original entry on oeis.org

1, 0, 1, 0, 48, 1, 0, 7584, 240, 1, 0, 2515968, 97664, 672, 1, 0, 1432498176, 63221760, 560448, 1440, 1, 0, 1247557386240, 60299053056, 628024320, 2141568, 2640, 1, 0, 1542446268088320, 79885647249408, 933093697536, 3819239424, 6374368, 4368, 1

Offset: 0

Stefano Spezia, Apr 30 2023

			The triangle begins:
    1;
    0,          1;
    0,         48,        1;
    0,       7584,      240,      1;
    0,    2515968,    97664,    672,    1;
    0, 1432498176, 63221760, 560448, 1440, 1;
    ...

Christian Krattenthaler and Thomas W. Müller, The congruence properties of Romik's sequence of Taylor coefficients of Jacobi's theta function theta_3, arXiv:2304.11471 [math.NT], 2023. See p. 6.

Cf. A317651, A362713, A362714.

U[x_]:=x*Hypergeometric2F1[3/4, 3/4, 3/2, 4*x^2]/Hypergeometric2F1[1/4, 1/4, 1/2, 4*x^2]; T[n_,k_]:=2^(n-k)(2n)!/(2k)!SeriesCoefficient[U[x]^(2k),{x,0,2n}]; Table[T[n,k],{n,0,7},{k,0,n}]//Flatten

cp's OEIS Frontend

A362714 a(0) = 1 and a(n) = 2^(n-1)Product_{j=1..n} (4j - 3)^2 - Sum_{m=1..n-1} binomial(2n, 2m)a(m)a(n-m)/2 for n > 0.

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A362715 Triangle read by rows: T(n, k) = 2^(n-k)(2n)!/(2k)! [x^(2n)] U[x]^(2k), where U(x) = x2F1([3/4, 3/4], [3/2], 4x^2)/2F1([1/4, 1/4], [1/2], 4*x^2).

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cp's OEIS Frontend

A362714 a(0) = 1 and a(n) = 2^(n-1)*Product_{j=1..n} (4*j - 3)^2 - Sum_{m=1..n-1} binomial(2*n, 2*m)*a(m)*a(n-m)/2 for n > 0.

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Formula

A362715 Triangle read by rows: T(n, k) = 2^(n-k)*(2*n)!/(2*k)! * [x^(2*n)] U[x]^(2*k), where U(x) = x*2F1([3/4, 3/4], [3/2], 4*x^2)/2F1([1/4, 1/4], [1/2], 4*x^2).

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A362714 a(0) = 1 and a(n) = 2^(n-1)Product_{j=1..n} (4j - 3)^2 - Sum_{m=1..n-1} binomial(2n, 2m)a(m)a(n-m)/2 for n > 0.

A362715 Triangle read by rows: T(n, k) = 2^(n-k)(2n)!/(2k)! [x^(2n)] U[x]^(2k), where U(x) = x2F1([3/4, 3/4], [3/2], 4x^2)/2F1([1/4, 1/4], [1/2], 4*x^2).