A268549 -id:A268549 - cp's OEIS frontend

A276013 Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 8 x^2 y) * (1 - u - z) * (1 - v - w)).

1, 12, 864, 100800, 14112000, 2139830784, 338341183488, 54913641209856, 9080061146956800, 1523231914913280000, 258557709598427086848, 44324863067728222027776, 7663322563977594870300672, 1334677098876385703362560000, 233951210561895726160281600000

Offset: 0

Gheorghe Coserea, Aug 16 2016

nonn

"The corresponding (order-five) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(5,C)." (see A. Bostan link).

			1 + 12*x + 864*x^2 + 100800*x^3 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..33
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (36).
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A004987, A268549, A268545-A268555.

diag_coeff := proc(expr, n)
    local var := [seq(indets(expr))], nvar := numelems(var);
    coeftayl(expr, var=[seq(0, i=1..nvar)], [seq(n, i=1..nvar)])
end proc:
pxy := (1 - 3*y - 2*x + 3*y^2 + 9*x^2*y):
expr := (1 - 9*x*y)/(pxy * (1-u-z-u*z) * (1-v-w)):
[seq(diag_coeff(expr, i), i=0..14)];

f = (1 - 9 x y)/((1 - 3y - 2x + 3 y^2 + 8 x^2 y)*(1 - u - z)*(1 - v - w));
a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}] &, f, {x, y, z, u, v, w}];
Array[a, 40, 0] (* Jean-François Alcover, Dec 03 2017 *)

a(n) = [(xyzuvw)^n] (1-9*x*y)/((1 - 3*y - 2*x + 3*y^2 + 8*x^2*y) * (1-u-z) * (1-v-w)).
From Vaclav Kotesovec, Dec 03 2017: (Start)
Recurrence: (n-1)^2*n^3*(3*n - 5)*a(n) = 24*(n-1)^2*(2*n - 1)^2*(3*n - 4)*(3*n - 2)*a(n-1) - 384*(2*n - 3)^2*(2*n - 1)^2*(3*n - 5)*(3*n - 2)*a(n-2).
a(n) ~ Gamma(1/3) * 2^(6*n - 7/3) * 3^(n + 1/2) / (Pi^2 * n^(4/3)). (End)

A276014 Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - v - z - w)).

Original entry on oeis.org

1, 72, 45360, 46569600, 59594535000, 86482063571904, 136141986298526208, 226888189910421811200, 394399917777684601926000, 708188604075430924446000000, 1304782547573305224852017990400, 2454776409299366206456800694732800, 4699106882676505497505898579906736000, 9127695522416954472516114289988092800000

Offset: 0

Gheorghe Coserea, Aug 16 2016

nonn

"The corresponding (order-four) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(4,C)." (see A. Bostan link).

			1 + 72*x + 45360*x^2 + 46569600*x^3 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..33
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (37).
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A004987, A268549, A268545-A268555.

diag_coeff := proc(expr, n)
    local var := [seq(indets(expr))], nvar := numelems(var);
    coeftayl(expr, var=[seq(0, i=1..nvar)], [seq(n, i=1..nvar)]);
end proc:
expr := (1-9*x*y)/((1-3*y-2*x + 3*y^2 + 9*x^2*y) * (1-u-v-z-w)):
[seq(diag_coeff(expr, i), i=0..14)];

f = (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y)*(1 - u - v - z - w));
a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}] &, f, {x, y, z, u, v, w}];
Array[a, 40, 0] (* Jean-François Alcover, Dec 03 2017 *)
Join[{1}, Table[FullSimplify[(4 * 9^n * Gamma[4*n] * Gamma[1/3 + n]) / (Gamma[1/3] * Gamma[n] * Gamma[1 + n]^4)], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 03 2017 *)

a(n) = [(xyzuvw)^n] (1-9*x*y)/((1-3*y-2*x + 3*y^2 + 9*x^2*y) * (1-u-v-z-w)).
From Vaclav Kotesovec, Dec 03 2017: (Start)
Recurrence: n^4*a(n) = 24*(2*n - 1)*(3*n - 2)*(4*n - 3)*(4*n - 1)*a(n-1).
For n > 0, a(n) = 4 * 9^n * Gamma(4*n) * Gamma(n + 1/3) / (Gamma(1/3) * Gamma(n) * Gamma(n+1)^4).
a(n) ~ 2^(8*n - 1/2) * 3^(2*n) / (Pi^(3/2) * Gamma(1/3) * n^(13/6)). (End)

A276015 Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - z - u z) * (1 - v - w)).

Original entry on oeis.org

1, 18, 1404, 158760, 21234150, 3126159036, 489778537248, 80153987120064, 13547671656870780, 2347445149320843000, 414851046001557525360, 74499573518808987538080, 13557818392046546526712440, 2495117936356342079352318000, 463604343771018075763879080000, 86854813070150110063356637257600

Offset: 0

Gheorghe Coserea, Aug 16 2016

nonn

"The corresponding (order-four) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(4,C)." (see A. Bostan link).

			1 + 18*x + 1404*x^2 + 158760*x^3 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..22
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (C.1).
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A004987, A268549, A268545-A268555.

diag_coeff := proc(expr, n)
    local var := [seq(indets(expr))], nvar := numelems(var);
    coeftayl(expr, var=[seq(0, i=1..nvar)], [seq(n, i=1..nvar)]);
end proc:
pxy := (1 - 3*y - 2*x + 3*y^2 + 9*x^2*y):
expr := (1 - 9*x*y)/(pxy * (1-u-z-u*z) * (1-v-w)):
[seq(diag_coeff(expr, i), i=0..14)];

f = (1-9x y)/((1 - 3y - 2x + 3y^2 + 9x^2 y)*(1 - u - z - u z)*(1 - v - w));
a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}]&, f, {x, y, z, u, v, w}];
Array[a, 40, 0] (* Jean-François Alcover, Dec 03 2017 *)

a(n) = [(xyzuvw)^n] (1-9*x*y)/((1 - 3*y - 2*x + 3*y^2 + 9*x^2*y) * (1-u-z-u*z) * (1-v-w)).
From Vaclav Kotesovec, Dec 03 2017: (Start)
Recurrence: (n-1)*n^3*a(n) = 18*(n-1)*(2*n - 1)^2*(3*n - 2)*a(n-1) - 36*(2*n - 3)*(2*n - 1)*(3*n - 5)*(3*n - 2)*a(n-2).
a(n) ~ Pi * 2^(2*n - 5/4) * 3^(2*n) * (1 + sqrt(2))^(2*n + 1) / (Gamma(1/3) * Pi^2 * n^(5/3)). (End)

A276016 Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - v - w)).

Original entry on oeis.org

1, 18, 1620, 211680, 32744250, 5578048476, 1011486123648, 191622065057280, 37496545808513220, 7522455177633825000, 1539274868268143806800, 320076654200479057190400, 67449486609121784655852000, 14374016718897406159444968000

Offset: 0

Gheorghe Coserea, Aug 16 2016

nonn

"The corresponding (order-three) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(3,C)." - (see A. Bostan link).

			1 + 18*x + 1620*x^2 + 211680*x^3 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..33
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (C.2).
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A004987, A268549, A268545-A268555.

diag_coeff := proc(expr, n)
    local var := [seq(indets(expr))], nvar := numelems(var);
    coeftayl(expr, var=[seq(0, i=1..nvar)], [seq(n, i=1..nvar)]);
end proc:
pxy := (1 - 3*y - 2*x + 3*y^2 + 9*x^2*y):
expr := (1 - 9*x*y)/(pxy * (1 - u - v - w)):
[seq(diag_coeff(expr, i), i=0..14)];

f = (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y)*(1 - u - v - w));
a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}]&, f, {x, y, u, v, w}];
Array[a, 40, 0] (* Jean-François Alcover, Dec 03 2017 *)
Join[{1}, Table[FullSimplify[3^(2*n) * Gamma[3*n] * Gamma[n + 1/3] / (Gamma[4/3] * Gamma[n] * Gamma[n + 1]^3)], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 03 2017 *)

a(n) = [(xyuvw)^n] (1-9*x*y)/((1-3*y-2*x+3*y^2+9*x^2*y) * (1-u-v-w)).
From Vaclav Kotesovec, Dec 03 2017: (Start)
Recurrence: n^3*a(n) = 9*(3*n - 2)^2*(3*n - 1)*a(n-1).
For n > 0, a(n) = 3^(2*n) * Gamma(3*n) * Gamma(n + 1/3) / (Gamma(4/3) * Gamma(n) * Gamma(n+1)^3).
a(n) ~ 3^(5*n + 1/2) / (2*Pi * Gamma(1/3) * n^(5/3)). (End)

A276017 Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 8 x^2 y) * (1 - u - v - w)).

Original entry on oeis.org

1, 18, 2160, 423360, 99792000, 25499650176, 6797581959168, 1860535606026240, 518890571236477440, 146835076503772800000, 42046646730013562757120, 12160617341681775057960960, 3547136319516173918512742400, 1042325945372157283978798694400, 308269259704090665806809006080000

Offset: 0

Gheorghe Coserea, Aug 16 2016

nonn

"The corresponding (order-five) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(5,C)." (see A. Bostan link).

			1 + 18*x + 2160*x^2 + 423360*x^3 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..33
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (C.3).
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A004987, A268549, A268545-A268555.

diag_coeff := proc(expr, n)
    local var := [seq(indets(expr))], nvar := numelems(var);
    coeftayl(expr, var=[seq(0, i=1..nvar)], [seq(n, i=1..nvar)]);
end proc:
pxy := (1 - 3*y - 2*x + 3*y^2 + 8*x^2*y):
expr := (1 - 9*x*y)/(pxy * (1 - u - v - w)):
[seq(diag_coeff(expr, i), i=0..14)];

f = (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 8 x^2 y)*(1 - u - v - w));
a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}]&, f, {x, y, u, v, w}];
Array[a, 40, 0] (* Jean-François Alcover, Dec 03 2017 *)

a(n) = [(xyuvw)^n] (1 - 9*x*y)/((1 - 3*y - 2*x + 3*y^2 + 8*x^2*y) * (1 - u - v - w)).
From Vaclav Kotesovec, Dec 03 2017: (Start)
Recurrence: (n-1)^2*n^3*(3*n - 5)*a(n) = 18*(n-1)^2*(3*n - 4)*(3*n - 2)^2*(3*n - 1)*a(n-1) - 216*(3*n - 5)^2*(3*n - 4)*(3*n - 2)^2*(3*n - 1)*a(n-2).
a(n) ~ Gamma(1/3) * 2^(2*n - 10/3) * 3^(4*n + 1) / (Pi^2 * n^(4/3)). (End)

cp's OEIS Frontend

A276013 Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 8 x^2 y) * (1 - u - z) * (1 - v - w)).

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A276014 Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - v - z - w)).

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A276015 Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - z - u z) * (1 - v - w)).

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A276016 Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - v - w)).

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A276017 Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 8 x^2 y) * (1 - u - v - w)).

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Programs

Maple

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