Real-rooted atlas

A075505 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-40\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-200\,x + 40\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-100\,x^{2} + 40\,x^{3}\right)\,P_{n-2}''(x) + \left(10 + 4\,x\right)\,P_{n-1}(x) + \left(20\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 100 + 120\,x + 9\,x^{2}\)
\(P_{1}(x) = 1000 + 2800\,x + 540\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01001209229
110002800540164356
210000600002250016002594125
310000012400008100001040003750362257786
410000002520000027090000560000035000075604959247609
5100000005080000008694000002721600002625000095760013720641686781384
61000000001020000000027225000000124320000001737750000952560002263800230408151792292921
7100000000020440000000083970000000054568000000010631250000082177200002881200004800000364501001705603176550
8100000000004092000000000256509000000002332000000000061682500000006461532000003135363000076032000093555005500012159919426560621
910000000000081880000000000778734000000000978401600000000344850000000000476514720000003074240400000101777280000180427500017050000798601442234794911085004
101000000000000163800000000000023546250000000000405204800000000001877125250000000033556723200000002800557760000001215751680000029119662000039325000002941510011232016988125163271447589
11100000000000003276400000000000071007300000000000016626792000000000001001875875000000000228370942800000000241713472000000001338388480000000415945530000007527520000007993986000484848001537901963661325108827624786
12100000000000000655320000000000000213759090000000000006776952000000000000052691730000000000000151449578280000000002002832531700000000138641817600000000543740769000000012662650000000017901683800001533168000076895000205800225159784577017067161025
131000000000000000131068000000000000006427517400000000000002748782416000000000000274047637500000000000098457356088000000000016081224759600000000013707289939200000000664835242500000000193754990000000003501365868000003970249920000279897800001179920002700002567304895994291644762256
1410000000000000000262140000000000000000193120425000000000000001110939664000000000000001413187912750000000000006302069963280000000000012596971345140000000000130662368179200000000077183460654300000000275833415000000000061959378338000000893121465600000829820082000048980400000175950000348160289348991249490383461118449
Technical details
Node id
n-65ef9e4a573f76ee · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-eeb15c314e2d2fa8

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:          100,            120,               9;
  1:         1000,           2800,             540,              16;
  2:        10000,          60000,           22500,            1600,              25;
  3:       100000,        1240000,          810000,          104000,            3750,             36;
  4:      1000000,       25200000,        27090000,         5600000,          350000,           7560,            49;
  5:     10000000,      508000000,       869400000,       272160000,        26250000,         957600,         13720,           64;
  6:    100000000,    10200000000,     27225000000,     12432000000,      1737750000,       95256000,       2263800,        23040,         81;
  7:   1000000000,   204400000000,    839700000000,    545680000000,    106312500000,     8217720000,     288120000,      4800000,      36450,      100;
  8:  10000000000,  4092000000000,  25650900000000,  23320000000000,   6168250000000,   646153200000,   31353630000,    760320000,    9355500,    55000,   121;
  9: 100000000000, 81880000000000, 778734000000000, 978401600000000, 344850000000000, 47651472000000, 3074240400000, 101777280000, 1804275000, 17050000, 79860, 144;
%F Formula
The row polynomials satisfy P(n,x) = -(40*x-9*x^2)*P(n-2,x) - (200*x-40*x^2-4*x^3)*d/dx P(n-2,x) - (100*x^2-40*x^3)*d^2/dx^2 P(n-2,x) + (10+4*x)*P(n-1,x) + (20*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 100 + 120*x + 9*x^2, P(1,x) = 1000 + 2800*x + 540*x^2 + 16*x^3.
%t Mathematica
P[0, x_] := 100 + 120*x + 9*x^2;
P[1, x_] := 1000 + 2800*x + 540*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(40*x - 9*x^2)*P[n - 2, x] - (200*x - 40*x^2 - 4*x^3)*D[P[n - 2, x], x] - (100*x^2 - 40*x^3)*D[P[n - 2, x], {x, 2}] + (10 + 4*x)*P[n - 1, x] + (20*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] = 100 + 120*x + 9*x^2; v[2] = 1000 + 2800*x + 540*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(40*x - 9*x^2)*v[i - 2] - (200*x - 40*x^2 - 4*x^3)*deriv(v[i - 2], x) - (100*x^2 - 40*x^3)*deriv(deriv(v[i - 2], x), x) + (10 + 4*x)*v[i - 1] + (20*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.