Real-rooted atlas

A094638 → D+1 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-16\,n x^{2} + 8\,n^{2} x + 32\,n^{2} x^{2} - n^{3} - 8\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 8\,n x + 2\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
213443289
3704167523841622
444639361227215488614438286
53276390561786723850883773441228801106316
627252413360253403279518081322598410653696294912037755252
7253296469187236389504152160768366527744499167232342540288825753601484306064
826022245714755253908492828401464329078945024178383298562064048128012365070336264241152066004219152
929288160745771968831558432053120976640213283584768554394633216922835124224930430255104495348350976951268147203273630384096
10358457760104013924481341649968001010678319360490844406092815985964583936351187490979845082617996902445479172243456218115971481603805072588800179090782858656
1147405779201546046530562268440258688197314209387521127886150487044434634692116481218264267468800232248163621273629779747172515842398110816927744104722694406144016742319390720010709892866518272
12673755321602441963893248402121470562563972867816309762618244074948096120977422271528964005881401623756895345539933634560160789382767575040185291162026770432135765027415130112544524731626291208036313307545600694894707199736064
1310242087206404087074476851274708379727948882738816506767366185822279436902432911170697667379212790233420751134723664551449158238208770543766424217190411663499170969288704122160148522708500488216581981983473664304850906376044544041788829199237120048611538606624440832
1416583626886400722890563932160145344521116500481785045703631247361494485404507584512901092842430838374440280520091486982144135302957418942808064342093616997920145408644936532947862814720886319957241466191872851382080383708299264529521436554474553344182835268038363709440234017443515727872003646773306351279853056
Technical details
Node id
n-401d70b5bb38eaa4 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-fb89365bfc869b8b

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 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:        3,         4;
  2:       13,        44,         32;
  3:       70,       416,        752,         384;
  4:      446,      3936,      12272,       15488,         6144;
  5:     3276,     39056,     178672,      385088,       377344,       122880;
  6:    27252,    413360,    2534032,     7951808,     13225984,     10653696,      2949120;
  7:   253296,   4691872,   36389504,   152160768,    366527744,    499167232,    342540288,     82575360;
  8:  2602224,  57147552,  539084928,  2840146432,   9078945024,  17838329856,  20640481280,  12365070336,   2642411520;
  9: 29288160, 745771968, 8315584320, 53120976640, 213283584768, 554394633216, 922835124224, 930430255104, 495348350976, 95126814720;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(8*n^3-8*n^2)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (2*n^2-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*P[n - 1, x])/(n - 1)]];
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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*v[i - 1])/(n - 1)); 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.