Real-rooted atlas

A075505 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-200 + 80\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-100\,x + 90\,x^{2}\right)\,P_{n-2}'(x) + \left(30 - 8\,x\right)\,P_{n-1}(x) + \left(10\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 130 + 32\,x + x^{2}\)
\(P_{1}(x) = 1700 + 820\,x + 63\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0130321163
117008206312584
225000200002800104147905
3410000490000109500710015511016756
4730000012320000406000040600015050216124101267
5137000000320200000147630000212100001183000282802871627251568
6265000000086000000005356000000105504000082740000292320048720368117746752289
7521000000002377000000001956150000005111500000053938500002635500006405000786004591542193884060
810330000000006723200000000722260000000024444200000003364735000002178792000072303000012804000120450560117782217375011
920570000000000193522000000000269976300000000116326100000000204122600000001700089600000738714900001768470000238095001771006711622582413547272
10410500000000000564200000000000010213840000000000553593040000000012169157000000001275056640000007045152400000218721360000394894500041756000251680792123153959628713473
118201000000000000165985000000000000390649350000000000264217590000000000717932215000000009303394100000000639679040000000250204240000005812606800008188180000697697003476209231910814844582978244
12163930000000000000491403200000000000015081886000000000000126662718000000000004211135565000000000665938072800000000560380821000000002703306320000000785248750000001414212800000159659500001119300004686501064137762014781386149715
1332777000000000000001461104200000000000005868135630000000000006102751210000000000002463653829000000000004704056120400000000047811174411000000002798391453000000009951637410000000222691898000000319697378000029558620000173446000618800121511644953882246414466016
146554500000000000000043571000000000000000002297590120000000000000029554379504000000000000144056266000000000000003293061276960000000000399993013056000000000280390528704000000001201132303800000000327039512800000005802855344000006791453760000523504800002608480008024001376175080880070271080291777
Technical details
Node id
n-5a9335c585a9eb7b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2d0fde8c3e0130d9

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:            130,              32,               1;
  1:           1700,             820,              63,               1;
  2:          25000,           20000,            2800,             104,              1;
  3:         410000,          490000,          109500,            7100,            155,             1;
  4:        7300000,        12320000,         4060000,          406000,          15050,           216,           1;
  5:      137000000,       320200000,       147630000,        21210000,        1183000,         28280,         287,          1;
  6:     2650000000,      8600000000,      5356000000,      1055040000,       82740000,       2923200,       48720,        368,        1;
  7:    52100000000,    237700000000,    195615000000,     51115000000,     5393850000,     263550000,     6405000,      78600,      459,      1;
  8:  1033000000000,   6723200000000,   7222600000000,   2444420000000,   336473500000,   21787920000,   723030000,   12804000,   120450,    560,   1;
  9: 20570000000000, 193522000000000, 269976300000000, 116326100000000, 20412260000000, 1700089600000, 73871490000, 1768470000, 23809500, 177100, 671, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(200-80*x-9*x^2)*P(n-2,x) - (100*x-90*x^2)*d/dx P(n-2,x) + (30-8*x)*P(n-1,x) + 10*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 130 + 32*x + x^2, P(1,x) = 1700 + 820*x + 63*x^2 + x^3.
%t Mathematica
P[0, x_] := 130 + 32*x + x^2;
P[1, x_] := 1700 + 820*x + 63*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(200 - 80*x - 9*x^2)*P[n - 2, x] - (100*x - 90*x^2)*D[P[n - 2, x], x] + (30 - 8*x)*P[n - 1, x] + 10*x*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] = 130 + 32*x + x^2; v[2] = 1700 + 820*x + 63*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(200 - 80*x - 9*x^2)*v[i - 2] - (100*x - 90*x^2)*deriv(v[i - 2], x) + (30 - 8*x)*v[i - 1] + 10*x*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.