Real-rooted atlas

A085880 → basis A013611 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n - 8\,n x + 4\,n x^{2} + 6\,n^{2} x - 8\,n^{2} x^{2} + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-12 + 48\,x^{2} + 8\,n - 72\,n x - 176\,n x^{2} + 48\,n^{2} x + 48\,n^{2} x^{2} - 96\,n^{2} x^{3} + 32\,n^{3} x^{2} + 64\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
221618
38128288424
4409604320512010440
522471684838411468889600260064
61344537604838401720320268800015482886495552
78448405504456192021626880506880005839257626492928162176256
854912307507241513472246005760768768000132843110412054282244498391044043115648
9366080234291203690086402624061440102502400002361655296032144752640239914188807591034880100610864640
1024893441792327683226189824267654266881254629376003613332602886557529538567341374177284645713346561274544128002498885655552
1117199104137592832027862548480264178237440144472473600049929686876161132664193024016907407196160160488279244808805941248000213103778201661950983417856
121203937281059464806423837958144025427155353601589197209600064076431491072174430285725696325467588526080411919916728320339028738048000164089909215232355055551119361533202207100928
138520171528179364659220243927531522399280300032016869939609600077736681721036824688594287329285527941811273728874537669361664095971212001280006967509991292928301524098796748858978672102604837884002888761344
14608583680063292702720017089029734400222790313574400174054932480000090230076997632003275017609543680085551480412569600162414138595737600222790313574400000215661023540019200139993331584204800547659098095616009771614312857600934702063305523200
1543818024960490761879552014354784976896020415694189363201754473719398400010105768623734784041265221880250368012319413179410022402728557528408391680449145272165990400054346577932084838404703775941229281280276020185440190464098497872273604608016153076721254400023030541357736919040
Technical details
Node id
n-f33cd188dfa59125 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       2,        16;
   3:       8,       128,        288;
   4:      40,       960,       4320,        5120;
   5:     224,      7168,      48384,      114688,        89600;
   6:    1344,     53760,     483840,     1720320,      2688000,      1548288;
   7:    8448,    405504,    4561920,    21626880,     50688000,     58392576,     26492928;
   8:   54912,   3075072,   41513472,   246005760,    768768000,   1328431104,   1205428224,    449839104;
   9:  366080,  23429120,  369008640,  2624061440,  10250240000,  23616552960,  32144752640,  23991418880,   7591034880;
  10: 2489344, 179232768, 3226189824, 26765426688, 125462937600, 361333260288, 655752953856, 734137417728, 464571334656, 127454412800;
%F Formula
The row polynomials satisfy (n+(6*n^2-8*n)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n,x) = (8*n-12+(48*n^2-72*n)*x+(32*n^3+48*n^2-176*n+48)*x^2+(64*n^3-96*n^2)*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 12 + (48*n^2 - 72*n)*x + (32*n^3 + 48*n^2 - 176*n + 48)*x^2 + (64*n^3 - 96*n^2)*x^3)*P[n - 1, x])/(n + (6*n^2 - 8*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 12 + (48*n^2 - 72*n)*x + (32*n^3 + 48*n^2 - 176*n + 48)*x^2 + (64*n^3 - 96*n^2)*x^3)*v[i - 1])/(n + (6*n^2 - 8*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.