Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A008277 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2\,x^{2} - 2\,x^{3} - x^{4}\right)\,P_{n-2}(x) + \left(-2\,x^{2} - 3\,x^{3} - 2\,x^{4}\right)\,P_{n-2}'(x) + \left(-x^{3} - x^{4}\right)\,P_{n-2}''(x) + \left(1 + 6\,x + 2\,x^{2}\right)\,P_{n-1}(x) + \left(3\,x + 2\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 6\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
116310
213075405151
31126903140070012673263
4151090753108034755158763234288994828
51204685503583000123365010769224479099504010395550113535027
618190784875101301203754250555927872400079681519689632355183932502674193613163254885
713276671253031694238001053834600252415963828611893311733022720604156410126626500162742581277640591501470159097183602
811310706437347527773491602826375825510503449938817995673350216332796022485759400727275833415005632670758744267888638323143498600117300217617599211936355
9152428658034430345038667800737946423550415848961067410450472443873136780080278401030153164124547729703378501423545395701281603415600376017090803410085420208751670843417621414930781945891416030315
1012097150522703087572603628020018956312842005159514076776824576176700504636106008812277667211094522881868367118713229127502950953685790118199650436213615663819838303209025606300019631193870012959111424594197583184218303664154200214031845922290572
111838860647059006503116653702990004820844409205005993819147900430306784939938211357793564015920320011088061930617210095934012973134600535067643984621002006595328180716052166732844380709608084612996101271910664399651219777803622484858880965142567936642151585380137191000595056556623401806863439720943
121335544304235939730751870851589552401218342487055375522215850200001458015948273809910141705521789768946948640105112930598367570451203163392175387500088259149488230892604347151449418496160148830595346867538036341354505254400064490919830208000842648258713536081707663368287058962214708650315816042382512440389500035389096057040990092575072002544992656191388756921
131134217726381259741590329973161541414003066910357920454508158807908030267306815330024103786907173810306545084506896009590694595248761601751431970705094236056750135871358836213101601586187799904738379120037968968464899042480011957608924522219980027519511068942723900470500905713319936060467840198668539058863449253789180435610126170219024492592624500010406845178730330342127830767813611512636000013893759126275614822356663874663066
1415368709103431438337607547992642977246280770100205586425350529776990233577583505641192884054971237629042579454891183423347452885076665634945289586554163920380750862110196250198655063485580665600065159920155835113454693640891158382691558173369453548264557792537434600010345393388633099301000225111920568129714432037091021745135330069046773562938773825700454731430551200985034220362321945800019948118085285390898107705838200310008028551008097190920001566378450021675162620224891136829775409212572949057258975
Technical details
Node id
n-5b410a2e2cae6c98 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-39851ca50dcc7dd5

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,      6,         3;
  2: 1,     30,        75,          40,            5;
  3: 1,    126,       903,        1400,          700,           126,              7;
  4: 1,    510,      9075,       31080,        34755,         15876,           3234,            288,              9;
  5: 1,   2046,     85503,      583000,      1233650,       1076922,         447909,          95040,          10395,           550,            11;
  6: 1,   8190,    784875,    10130120,     37542505,      55927872,       40007968,       15196896,        3235518,        393250,         26741,          936,          13;
  7: 1,  32766,   7125303,   169423800,   1053834600,    2524159638,     2861189331,     1733022720,      604156410,     126626500,      16274258,      1277640,       59150,       1470,        15;
  8: 1, 131070,  64373475,  2777349160,  28263758255,  105034499388,   179956733502,   163327960224,    85759400727,   27583341500,    5632670758,    744267888,    63832314,    3498600,    117300,    2176,     17;
  9: 1, 524286, 580344303, 45038667800, 737946423550, 4158489610674, 10450472443873, 13678008027840, 10301531641245, 4772970337850, 1423545395701, 281603415600, 37601709080, 3410085420, 208751670, 8434176, 214149, 3078, 19;
%F Formula
The row polynomials satisfy P(n,x) = -(2*x^2+2*x^3+x^4)*P(n-2,x) - (2*x^2+3*x^3+2*x^4)*d/dx P(n-2,x) - (x^3+x^4)*d^2/dx^2 P(n-2,x) + (1+6*x+2*x^2)*P(n-1,x) + (3*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 6*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(2*x^2 + 2*x^3 + x^4)*P[n - 2, x] - (2*x^2 + 3*x^3 + 2*x^4)*D[P[n - 2, x], x] - (x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (1 + 6*x + 2*x^2)*P[n - 1, x] + (3*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {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; v[2] = 1 + 6*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(2*x^2 + 2*x^3 + x^4)*v[i - 2] - (2*x^2 + 3*x^3 + 2*x^4)*deriv(v[i - 2], x) - (x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 6*x + 2*x^2)*v[i - 1] + (3*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.