Real-rooted atlas

A075503 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-32\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-128\,x + 32\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-64\,x^{2} + 32\,x^{3}\right)\,P_{n-2}''(x) + \left(8 + 4\,x\right)\,P_{n-1}(x) + \left(16\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 64 + 96\,x + 9\,x^{2}\)
\(P_{1}(x) = 512 + 1792\,x + 432\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
064969169
15121792432162752
24096307201440012802550521
332768507904414720665603000361024988
4262144825753611096064286720022400060484922713041
52097152133169152284884992111476736134400006128641097664545691936
616777216213909504071368704004073717760711782400487710721448832184328114128481233
713421772834292629504176097853440143046737920348364800003365978112147517440307200029160100391924515404
810737418245492189429764303506898944489055846400016169697280002117314805761284244684838928384059875204400012111586297018649
98589934592879179805491210451990819635216414854977945672320286720000124915474759681007367094272416879738889237888001091200063888144363330669994272
10687194767361407031286169602528259342336000543856676306944031492935778304007037354917232647341494134374439837751050241192741355522013440000188256648985616912038147045992809
1154975581388822515249357783046099480625545216017852881969348608013446950368051200038314286354792448506909891231744035085050970112013629703127040308327219200409292083231030272123032196419993382252274012
1243980465111043602659799570841614689412812363530245821357441312358400565773142799155200020327218303299747843360195399615774722907529650634752014253838014873604149297152000073325296844878498201604921280016464022515383345431305556353
133518437208883257644316011737907235335650595162816512188894889288637874176235405128121712640000105717781112146624512215838545069085818882299701640845852672139426055847936000507917080985600011473275676262416262143672321433076736075514880216000256589958105209939278016
1428147497671065692232312993664204808493526113941716992006107455391627812536320971135338938200227840054134368778783041781761352589498900635713536175372061921117208576129492359102469242885784645979340800001624227927503667229265804184780833989430558722507796480011260800027852828923632327582748018541601
Technical details
Node id
n-9773d624cfbe6ef6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-cfb8d8d9d8395f5d

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         64,            96,               9;
  1:        512,          1792,             432,              16;
  2:       4096,         30720,           14400,            1280,             25;
  3:      32768,        507904,          414720,           66560,           3000,             36;
  4:     262144,       8257536,        11096064,         2867200,         224000,           6048,            49;
  5:    2097152,     133169152,       284884992,       111476736,       13440000,         612864,         10976,          64;
  6:   16777216,    2139095040,      7136870400,      4073717760,      711782400,       48771072,       1448832,       18432,        81;
  7:  134217728,   34292629504,    176097853440,    143046737920,    34836480000,     3365978112,     147517440,     3072000,     29160,      100;
  8: 1073741824,  549218942976,   4303506898944,   4890558464000,  1616969728000,   211731480576,   12842446848,   389283840,   5987520,    44000,   121;
  9: 8589934592, 8791798054912, 104519908196352, 164148549779456, 72320286720000, 12491547475968, 1007367094272, 41687973888, 923788800, 10912000, 63888, 144;
%F Formula
The row polynomials satisfy P(n,x) = -(32*x-9*x^2)*P(n-2,x) - (128*x-32*x^2-4*x^3)*d/dx P(n-2,x) - (64*x^2-32*x^3)*d^2/dx^2 P(n-2,x) + (8+4*x)*P(n-1,x) + (16*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 64 + 96*x + 9*x^2, P(1,x) = 512 + 1792*x + 432*x^2 + 16*x^3.
%t Mathematica
P[0, x_] := 64 + 96*x + 9*x^2;
P[1, x_] := 512 + 1792*x + 432*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(32*x - 9*x^2)*P[n - 2, x] - (128*x - 32*x^2 - 4*x^3)*D[P[n - 2, x], x] - (64*x^2 - 32*x^3)*D[P[n - 2, x], {x, 2}] + (8 + 4*x)*P[n - 1, x] + (16*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 64 + 96*x + 9*x^2; v[2] = 512 + 1792*x + 432*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(32*x - 9*x^2)*v[i - 2] - (128*x - 32*x^2 - 4*x^3)*deriv(v[i - 2], x) - (64*x^2 - 32*x^3)*deriv(deriv(v[i - 2], x), x) + (8 + 4*x)*v[i - 1] + (16*x - 4*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

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.