Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 - 2\,x + 4\,n x\right)\,P_{n}(x) = \left(1 + 6\,x + 12\,x^{2} + 8\,x^{3} + 4\,n x + 16\,n x^{2} + 16\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1181221
21167212880297
3124180640120011524483645
41323361792560010752125448192230441553
514054038401680048384940801228801036805120011264452709
614879270403960015206441395281100811404801126400743424294912532484782969
715610921164880080384384134534435143686918912102502401127526489456644845568160563224576049424013
816414401792014560083865635875841171456029652480585728009020211210734796896911360642252802949120083886081114112502211745
917218362611224480016450568316672325877761008184322489344004928901127821066249884958729826467847520256004278190081704591364246732849807365036466357
1018022803648038760029767681736448079380480290234880859955200208109158441277849606707650560889061376095256576008128561152539787264026895974409463398402097152002202009649977243081
11188277249280585200505612833426624174637056736750080254679040072838205441733669683234432606208570481049607858667520089414172672831272386566212970086436434083840161480704005086642176101502156896468992491636600541
12196331264768850080816076860299008354410496169452518466944204802209158758461345234944143990898688286277763072481998274560685508657152819397066752816561782784670387142656445686743040233985540096933819842562662544179248318382084194304004801302120177
13110439008320011960001262976010314304067358720035994816001599769600059831383040189878108160514253209600119282401280023734763520004050732974080591786770432073717383168007781279334400689753292800050696867020803034914488320144221143040052344913920013631488000022682796032181193932846600873519365
14111245361048321638000188697601687795201212456960716107392035363328000147818711040527749447680161989761024042941664460809859055616000196304751820803389324230656050663946977280653627464089607242409574400068440770478080546284607897603634372804608019786377461760858783744000028580323000326849130659841052266987527784628224449910251614233
Technical details
Node id
n-37dbd4914c5d1bbb · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-111fd8abade84f64

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  8,   12;
  2: 1, 16,   72,   128,     80;
  3: 1, 24,  180,   640,   1200,    1152,     448;
  4: 1, 32,  336,  1792,   5600,   10752,   12544,     8192,      2304;
  5: 1, 40,  540,  3840,  16800,   48384,   94080,   122880,    103680,     51200,     11264;
  6: 1, 48,  792,  7040,  39600,  152064,  413952,   811008,   1140480,   1126400,    743424,    294912,     53248;
  7: 1, 56, 1092, 11648,  80080,  384384, 1345344,  3514368,   6918912,  10250240,  11275264,   8945664,   4845568,   1605632,    245760;
  8: 1, 64, 1440, 17920, 145600,  838656, 3587584, 11714560,  29652480,  58572800,  90202112, 107347968,  96911360,  64225280,  29491200,   8388608,   1114112;
  9: 1, 72, 1836, 26112, 244800, 1645056, 8316672, 32587776, 100818432, 248934400, 492890112, 782106624, 988495872, 982646784, 752025600, 427819008, 170459136, 42467328, 4980736;
%F Formula
The row polynomials satisfy (1+(4*n-2)*x)*P(n,x) = (1+(4*n+6)*x+(16*n+12)*x^2+(16*n+8)*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[((1 + (4*n + 6)*x + (16*n + 12)*x^2 + (16*n + 8)*x^3)*P[n - 1, x])/(1 + (4*n - 2)*x)]];
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] = ((1 + (4*n + 6)*x + (16*n + 12)*x^2 + (16*n + 8)*x^3)*v[i - 1])/(1 + (4*n - 2)*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.