Real-rooted atlas

A178820 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n - 2\,n x + 6\,n^{2} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(6 + 36\,x + 72\,x^{2} + 48\,x^{3} + 2\,n + 48\,n x + 144\,n x^{2} + 112\,n x^{3} + 12\,n^{2} x + 64\,n^{2} x^{2} + 80\,n^{2} x^{3} + 8\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
186472
24064014402120
31603840172802048041760
456017920120960286720224000650160
51792716806451202293760358400020643848660736
65376258048290304013762560322560003715891216859136103203072
71536086016011612160688128002150400003715891203371827201258291201130941440
842240270336042577920302776320118272000027249868803709009920276824064087588864011608945920
9112640811008014598144012111052805677056000163499212802967207936033218887680210213273605767168000113071749120
102928642342912047443968044983910402460057600085019590656192868515840287897026560273277255680149946368000362870210561054892906496
117454726560153614760345601574436864098402304000396758089728108006368870420152791859202550587719680209924915200010160365895682198486384649493512118272
12186368017891328044281036805248122880036900864000017003918131205400318443520120916751155201912940789760020992491520000152405488435206595459153920129008402432082866475560960
13458752047710208012881756160167939932160131203072000068015672524802468717002752064488933949440122428210544640167939932160000162565854330880105527346462720412826887782407365868912640704580906516480
1411141120124780544036498309120519087063040446090444800025694809620480104920472616960313231964897280693759859752960114199153868800013818097618124801195976593244160701805709230080250439543029760410706247680005855718626426880
Technical details
Node id
n-dc886079d2047822 · 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 0)
  • 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:      1;
  1:      8,      64;
  2:     40,     640,      1440;
  3:    160,    3840,     17280,      20480;
  4:    560,   17920,    120960,     286720,     224000;
  5:   1792,   71680,    645120,    2293760,    3584000,     2064384;
  6:   5376,  258048,   2903040,   13762560,   32256000,    37158912,    16859136;
  7:  15360,  860160,  11612160,   68812800,  215040000,   371589120,   337182720,   125829120;
  8:  42240, 2703360,  42577920,  302776320, 1182720000,  2724986880,  3709009920,  2768240640,   875888640;
  9: 112640, 8110080, 145981440, 1211105280, 5677056000, 16349921280, 29672079360, 33218887680, 21021327360, 5767168000;
%F Formula
The row polynomials satisfy (n+(6*n^2-2*n)*x+4*n^3*x^2)*P(n,x) = (2*n+6+(12*n^2+48*n+36)*x+(8*n^3+64*n^2+144*n+72)*x^2+(16*n^3+80*n^2+112*n+48)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 6 + (12*n^2 + 48*n + 36)*x + (8*n^3 + 64*n^2 + 144*n + 72)*x^2 + (16*n^3 + 80*n^2 + 112*n + 48)*x^3)*P[n - 1, x])/(n + (6*n^2 - 2*n)*x + 4*n^3*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 6 + (12*n^2 + 48*n + 36)*x + (8*n^3 + 64*n^2 + 144*n + 72)*x^2 + (16*n^3 + 80*n^2 + 112*n + 48)*x^3)*v[i - 1])/(n + (6*n^2 - 2*n)*x + 4*n^3*x^2)); 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

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.