Real-rooted atlas

A085880 → basis A013610 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-600 + 180\,x + 640\,n - 192\,n x - 160\,n^{2} + 48\,n^{2} x\right)\,P_{n-2}(x) + \left(42 + 18\,x - 70\,n - 30\,n x + 28\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2538
3507818146
462516659451353370
58750334743288610962113487206
6131250657594954450538650125874102062418024
72062500127959482504422820746044805107614434209622870239444
833515625248070537617176989690381549389084553114150465165752739382232109893214
9558593750480427233014584245390208471220101578450159066596458501564747470190772010938223065003282630
10949609375093064208094334596290886587661582222567688505670319793309550107171962326209580253742201071158956987462042726747776
1116402343750018042750116047511929455420157436663553001869866648348413390483669524595670947887616502233761002760473827802545586643699178700465222472304028
12287041015625035019124299978165907118566098405678177323874576103212736506506328539985450285181171727634104329325808162245845815529863590434252314295056633690104137635422109897566806484
13507841796875006805088598296523617845364543004101388088119495241684524313715770817693293110503220121983828588984605628405819339828174356183276679635701069419061246245120713820342692926405211054610529269003627264949468
1490686035156250013240167218649300780986643735609002472557120598141004723370274063969005806717775739765004776871623606225002690344430998965001047418937316441002808460361195010050800760806905005910268557933003987555940890011844225567002277770474706522800
15163234863281250002579111278023999601672176800184476040591029385023777868012798506145999256200181090618119361062001742306377026195540011685546195968791800553799112490654740018602947590479679604391562474980370007110183361674636075059226263192404647674112490801279176361236075789410635589368680
Technical details
Node id
n-7436bc4920832f70 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-0fee5d4c7376c95e

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:          5,           3;
   3:         50,          78,           18;
   4:        625,        1665,          945,          135;
   5:       8750,       33474,        32886,        10962,         1134;
   6:     131250,      657594,       954450,       538650,       125874,        10206;
   7:    2062500,    12795948,     25044228,     20746044,      8051076,      1443420,        96228;
   8:   33515625,   248070537,    617176989,    690381549,    389084553,    114150465,     16575273,      938223;
   9:  558593750,  4804272330,  14584245390,  20847122010,  15784501590,   6659645850,   1564747470,   190772010,    9382230;
  10: 9496093750, 93064208094, 334596290886, 587661582222, 567688505670, 319793309550, 107171962326, 20958025374, 2201071158, 95698746;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = -(160*n^2-640*n+600-(48*n^2-192*n+180)*x)*P(n-2,x) + (28*n^2-70*n+42+(12*n^2-30*n+18)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(160*n^2 - 640*n + 600 - (48*n^2 - 192*n + 180)*x)*P[n - 2, x] + (28*n^2 - 70*n + 42 + (12*n^2 - 30*n + 18)*x)*P[n - 1, x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 3*x; for(n = 3, N, my(i = n); v[i] = (-(160*n^2 - 640*n + 600 - (48*n^2 - 192*n + 180)*x)*v[i - 2] + (28*n^2 - 70*n + 42 + (12*n^2 - 30*n + 18)*x)*v[i - 1])/(n^2 - n)); 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

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

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.