A084768 -id:A084768 - cp's OEIS frontend

A299864 a(n) = (-1)^n*hypergeom([-n, n - 1/2], [1], 4).

1, 1, 19, 239, 3011, 38435, 496365, 6470385, 84975315, 1122708899, 14906800361, 198740733581, 2658870294349, 35677678567549, 479965685669059, 6471364940381007, 87425255326277907, 1183139999323074963, 16036589185819644633, 217668383345249016045

Offset: 0

Peter Luschny, Mar 16 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..875

Cf. A299507, A245926, A084768, A245927.

Cf. A300945, A300946.

seq((-1)^n*orthopoly[P](n,0,-3/2,-7),n=0..100); # Robert Israel, Mar 21 2018

a[n_] := (-1)^n Hypergeometric2F1[-n, n - 1/2, 1, 4]; Table[a[n], {n, 0, 19}]

From Robert Israel, Mar 21 2018: (Start)
a(n) = JacobiP(n,0,-3/2,-7).
n*(2*n-3)*(4*n-7)*a(n)+(2*n-5)*(n-1)*(4*n-3)*a(n-2)-(4*n-5)*(28*n^2-70*n+39)*a(n-1) = 0. (End)
a(n) ~ sqrt(3) * (1 + sqrt(3))^(4*n - 1)  / (2^(2*n + 1) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 05 2018

A384364 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..kn} 3^i Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 21, 9, 1, 1, 219, 657, 27, 1, 1, 3045, 119241, 22869, 81, 1, 1, 52923, 40365873, 80850987, 836001, 243, 1, 1, 1103781, 21955523049, 747786838869, 60579666801, 31436181, 729, 1, 1, 26857659, 17512689629457, 14298291269335467, 16117269494868801, 48066954848379, 1204022961, 2187, 1

Offset: 0

Seiichi Manyama, May 27 2025

			Square array begins:
  1,  1,      1,           1,                 1, ...
  1,  3,     21,         219,              3045, ...
  1,  9,    657,      119241,          40365873, ...
  1, 27,  22869,    80850987,      747786838869, ...
  1, 81, 836001, 60579666801, 16117269494868801, ...

Columns k=0..2 give A000012, A000244, 3^n * A084768(n).
Rows n=0..1 give A000012, A032033.
Cf. A262809, A384362.

a(n, k) = sum(i=0, k*n, 3^i*sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));

A(n,k) = (1/4) * Sum_{j>=0} (3/4)^j * binomial(j,n)^k.

A355951 Negated column 0 of the irregular triangle A355587.

Original entry on oeis.org

0, 0, 2, 24, 280, 3400, 212538, 2708944, 244962336, 3195918288, 42013225014, 111125508824, 11603576403816, 30966112647080, 188641282541015866, 2532986569522773024, 34096877865475065728, 459984329860282638816, 105694712757690117569946, 1431044069320995796765272, 73738714208458783084303688

Offset: 0

Hugo Pfoertner, Jul 21 2022

a(n) are the numerators u in the representation R = s/t - (2*sqrt(3)/Pi)*u/v of the resistance between two nodes with distance n on the same grid line in an infinite triangular lattice of one-ohm resistors. The corresponding denominators are A355952. s(n)/t(n) = (1/3)*Sum_{k=0..n-1} A084768(k-1) for n >= 0.

R(n) > 1 [ohm] for n >= 38. Cserti (2000, page 11) claims that R(n) is logarithmically divergent for large values of n.

J. Cserti, Application of the lattice Green's function for calculating the resistance of infinite networks of resistors, arXiv:cond-mat/9909120 [cond-mat.mes-hall], 1999-2000.

Cf. A355587, A355952 (denominators).

Cf. A084768, A355585, A355586, A355588.

Rtri(n, p) = {my(alphat(beta)=acosh(2/cos(beta)-cos(beta))); intnum (beta=0, Pi/2, (1 - exp (-abs(n-p) * alphat(beta))*cos((n+p)*beta)) / (cos(beta)*sinh(alphat(beta)))) / Pi};
D(n) = subst(pollegendre(n), x, 7);
uv(k) = (Rtri(k) - sum(j=0, k-1, D(j))/3) / (2*sqrt(3)/Pi);
poddpri(primax) = {my(pp=1,p=2); while (p<=primax, p=nextprime(p+1); pp*=p); pp};
for (k=0, 20, print1(-numerator(bestappr(uv(k),poddpri(k))), ", "))
\\ for A355952 replace by
\\ for (k=0, 20, print1(denominator(bestappr(uv(k),poddpri(k))),", "))

cp's OEIS Frontend

A299864 a(n) = (-1)^n*hypergeom([-n, n - 1/2], [1], 4).

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A384364 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..kn} 3^i Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.

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PARI

Formula

A355951 Negated column 0 of the irregular triangle A355587.

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Programs

PARI

cp's OEIS Frontend

A299864 a(n) = (-1)^n*hypergeom([-n, n - 1/2], [1], 4).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A384364 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..k*n} 3^i * Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A355951 Negated column 0 of the irregular triangle A355587.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A384364 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..kn} 3^i Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.