Real-rooted atlas

A085880 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-30 - 18\,x + 20\,n + 12\,n x\right)\,P_{n-1}(x) + \left(-18\,x + 12\,n x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2538
3507818146
4625193510801353775
587504859449518143641134122360
6131250124500620714401048950187110102064693962
7206250032540508833512686568452019887120240570096228206027844
8335156258661625833297077784380132052317391874503502699203064861893822310119120726
9558593750234328251301295544779102103657355801365671607304074320250058524265803877988409382230547471603250
1094960937506430704997865084352290160113348646882901003931239311041635228094468732404240569397616857248806360469569874632246811701962
1116402343750017870544238404199869935842044600887652295320707051852764920391705358908500112532157354708174907148083201462885830900611601985809917870042049097277466200
12287041015625050209748476906278826332444219363154212928881214248378905588613600348434195776266601317174493559324427732426997815543321354173237102221024346541076367599308010413763542139452162980320190
1350784179687500142436916151213323121462980442792441646225742286557432324500907785783929229794026909727080001441853695659240864394961466926969712635212414830525846040628151673500330348611314260950696505511211054610529210103795283299604124
1490686035156250040750673378583270012415501261901556000856557117362308635002146565412515819295002475824323322417067001503601538324309424005212342581099619680010810715727819407400137094223322396340010601459232507000048296803787037001180474481511001184422556700775494801285138442400
1516323486328125000117459880248288567604960549226961886107604451135285134572325200140620620113481074112002014422577343848772100015123711006513492463080650264231729949819840016915717207431050580002746882751278501836002816401475177762472018056578729174221606958435143710232014625249730131601279176361236062787894356105327727720
Technical details
Node id
n-21da7519920fe4af · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c377b8329dc9dd7c

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,         1935,          1080,            135;
   5:       8750,        48594,         49518,          14364,           1134;
   6:     131250,      1245006,       2071440,        1048950,         187110,         10206;
   7:    2062500,     32540508,      83351268,       65684520,       19887120,       2405700,        96228;
   8:   33515625,    866162583,    3297077784,     3801320523,     1739187450,     350269920,     30648618,      938223;
   9:  558593750,  23432825130,  129554477910,   210365735580,   136567160730,   40743202500,   5852426580,   387798840,    9382230;
  10: 9496093750, 643070499786, 5084352290160, 11334864688290, 10039312393110, 4163522809446, 873240424056, 93976168572, 4880636046, 95698746;
%F Formula
The row polynomials satisfy n*P(n,x) = (20*n-30+(12*n-18)*x)*P(n-1,x) + (12*n-18)*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((20*n - 30 + (12*n - 18)*x)*P[n - 1, x] + (12*n - 18)*x*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((20*n - 30 + (12*n - 18)*x)*v[i - 1] + (12*n - 18)*x*deriv(v[i - 1], x))/(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 (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Ray 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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.