Real-rooted atlas

A136572 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1189
223296130
3614486415362550
4247686912245763072063000
51204800576003072007680007372801875000
672034560518400368640013824000265420802064384065250000
7504028224050803204515840022579200065028096010115481606606028802598750000
840320258048054190080578027520361267200013872660480323695411204227858432023781703680116550000000
936288026127360627056640780337152058525286400280921374720873977610240171228266496019263179980809512681472005811750000000
103628800290304000783820800011147673600097542144000056184274944002184944025600057076088832000963158999040009512681472000041855798476800318937500000000
113991680035126784001053803520001686085632000168608563200001133049544704005287564541952001726551687168000388474129612800057551722905600005064551615692800200907832688640019100812500000000
1247900160045984153600151747706880026977370112000303495413760000233084477767680012690154900684800497246885904384001398506866606080002762482699468800003646477163298816002893072790716416001044720729980928001239603750000000000
136227020800647610163200233139658752004559175548928005698969436160000492390959284224003063765968879616001400578728630681600472695320912855040011671489405255680000205418213532499968002444646508155371520017655780336677683200585043608789319680086655318750000000000
148717829120097639686144003807947759616008123621887180800111699800948736000107231808910786560075062266237550592003921620440165908480015441380483153264640045752238468602265600010065492463092498432001597169051994842726400173026647299441295360011466854732270665728003510261652735918080006491547562500000000000
Technical details
Node id
n-746c648eef173fb9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-14ef8ed0e797c5a3

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      1,        8;
  2:      2,       32,        96;
  3:      6,      144,       864,       1536;
  4:     24,      768,      6912,      24576,       30720;
  5:    120,     4800,     57600,     307200,      768000,       737280;
  6:    720,    34560,    518400,    3686400,    13824000,     26542080,     20643840;
  7:   5040,   282240,   5080320,   45158400,   225792000,    650280960,   1011548160,     660602880;
  8:  40320,  2580480,  54190080,  578027520,  3612672000,  13872660480,  32369541120,   42278584320,   23781703680;
  9: 362880, 26127360, 627056640, 7803371520, 58525286400, 280921374720, 873977610240, 1712282664960, 1926317998080, 951268147200;
%F Formula
The row polynomials satisfy (1+4*n*x)*P(n,x) = (n+(4*n^2+8*n)*x+(16*n^2+16*n)*x^2)*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[((n + (4*n^2 + 8*n)*x + (16*n^2 + 16*n)*x^2)*P[n - 1, x])/(1 + 4*n*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] = ((n + (4*n^2 + 8*n)*x + (16*n^2 + 16*n)*x^2)*v[i - 1])/(1 + 4*n*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. [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.