A107657
Coefficients of 12th root of theta series of lattice in A004046.
Original entry on oeis.org
1, 0, 0, 2184, 44226, 530712, -22289904, -1041080040, -23414482728, 86664734520, 22704271546320, 824932708688088, 10338270616438674, -363177176817506688, -24534229526034608016, -614775613733783853624, -526997882017733986314, 591470477348411755418688, 24257417213770154760619728, 384176112414487265101313448
Offset: 0
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terms = 20; QP = QPochhammer; Q9 = (1 + (9*q*QP[q^9]^3)/QP[q]^3); s = (1/(3^(1/4)*QP[q^3]))*QP[q]^3*(-8 + 16*Q9^3 + 64*Q9^6 - 72*Q9^9 + 27*Q9^12)^(1/12) + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 06 2017, after Michael Somos *)
Original entry on oeis.org
1, 0, 0, 26208, 530712, 6368544, 47331648, 256864608, 1116087336
Offset: 0
A004010
Theta series of 12-dimensional Coxeter-Todd lattice K_12.
Original entry on oeis.org
1, 0, 756, 4032, 20412, 60480, 139860, 326592, 652428, 1020096, 2000376, 3132864, 4445532, 7185024, 10747296, 13148352, 21003948, 27506304, 33724404, 48009024, 64049832, 70709184, 102958128, 124782336, 142254252, 189423360, 237588120, 248250240, 344391264
Offset: 0
G.f. = 1 + 756*x^2 + 4032*x^3 + 20412*x^4 + 604890*x^5 + 139860*x^6 + ...
G.f. = 1 + 756*q^4 + 4032*q^6 + 20412*q^8 + 60480*q^10 + 139860*q^12 + 326592*q^14 + 652428*q^16 + 1020096*q^18 + 2000376*q^20 + ...
- J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 129.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..500 from N. J. A. Sloane)
- N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
- G. Nebe and N. J. A. Sloane, Home page for this lattice
- Eric Weisstein's World of Mathematics, Coxeter-Todd Lattice
- Eric Weisstein's World of Mathematics, Theta Series
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A := Basis( ModularForms( Gamma1(3), 6), 29); A[1] + 756*A[3]; /* Michael Somos, Dec 20 2015 */
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# Jacobi theta constants th2, th3: maxd := 201: temp0 := trunc(evalf(sqrt(maxd)))+2: a := 0: for i from -temp0 to temp0 do a := a+q^( (i+1/2)^2): od: th2 := series(a,q,maxd); a := 0: for i from -temp0 to temp0 do a := a+q^(i^2): od: th3 := series(a,q,maxd);
# get phi0 and phi1: phi0 := series( subs(q=q^2, th2)*subs(q=q^6, th2)+subs(q=q^2, th3)*subs(q=q^6, th3), q, maxd); phi1 := series( subs(q=q^2, th2)*subs(q=q^6, th3)+subs(q=q^2, th3)*subs(q=q^6, th2), q, maxd);
K_12 := series( subs(q=q^2,phi0)^6+45*subs(q=q^2,phi0)^2*subs(q=q^2,phi1)^4+18*subs(q=q^2,phi1)^6,q,maxd);
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maxd = 51; temp0 = Floor[ Sqrt[maxd] ]+2; a = 0; Do[ a=a+q^(i+1/2)^2, {i, -temp0, temp0}]; th2[q_] = Normal[ Series[a, {q, 0, maxd}]]; a = 0; Do[ a=a+q^i^2, {i, -temp0, temp0}]; th3[q_] = Normal[ Series[a, {q, 0, maxd}]]; phi0[q_] = Normal[ Series[ th2[q^2]*th2[q^6] + th3[q^2]*th3[q^6], {q, 0, maxd}]]; phi1[q_] = Normal[ Series[ th2[q^2]*th3[q^6] + th3[q^2]*th2[q^6], {q, 0, maxd}]]; K12 = Series[ phi0[q^2]^6 + 45*phi0[q^2]^2*phi1[q^2]^4 + 18*phi1[q^2]^6, {q, 0, maxd}]; CoefficientList[ K12, q^2 ] (* Jean-François Alcover, Nov 28 2011, translated from Maple *)
a[ n_] := With[ {U1 = QPochhammer[ q]^3, U3 = QPochhammer[ q^3]^3, U9 = QPochhammer[ q^9]^3}, With[ {z = (1 + 9 q U9/U1)^3}, SeriesCoefficient[ (U1^3/U3)^2 (3 z^2 - 4 z + 4) / 3, {q, 0, n}]]]; (* Michael Somos, Dec 25 2015 *)
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{a(n) = my(A, U1, U3, U9, z); if( n<0, 0, A = x * O(x^n); U1 = eta(x + A)^3; U3 = eta(x^3 + A)^3; U9 = eta(x^9 + A)^3; z = (1 + 9 * x * U9/U1)^3; polcoeff( (U1^3/U3)^2 * (3*z^2 - 4*z +4) / 3, n))}; /* Michael Somos, Dec 25 2015 */
A071687
Non-palindromic numbers such that either x=q1.Rev[x] or Rev[x]=q2.x, where R[x]=A004086[x] and q1 or q2 are integers not divisible by 10.
Original entry on oeis.org
510, 540, 810, 1089, 2100, 2178, 4200, 5200, 5610, 5700, 5940, 6300, 8400, 8712, 8910, 9801, 10989, 21978, 23100, 27000, 46200, 51510, 52200, 52800, 54540, 56610, 57200, 59940, 65340, 69300, 81810, 87912, 89910, 98901, 109989, 212100, 217800
Offset: 1
Includes special cases of A071685. Examples represented by {n, Rev[n], integer-quotient} triples: {1089, 9801, 9}, {87912, 21979, 4}, {5610, 165, 34}, {610000, 16, 38125}, etc.
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nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] ed[x_] := IntegerDigits[x] red[x_] := Reverse[ed[x]] Do[s=Mod[ma=Max[{n, tn[red[n]]}], mi=Min[{n, r=tn[red[n]]}]]; If[Equal[s, 0]&&!Equal[n, r] &&!Equal[Mod[ma/mi, 10], 0], Print[{n, r, Max[r/n, n/r]}]], {n, 1, 1000000}]
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