Real-rooted atlas

A049374 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 6\,n x\right)\,P_{n-1}(x) + \left(-5\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1167
21184261
3136276336649
4160960420030248245
51902460234006602430240122215
6112652508736055742410866243326402069425
71168991225536029161441334995218805248399168039328465
8121617136632016113430249500198432585414434248614451891840827226505
912702772013860003599164848066480030809116808162985600655912857672648576019047582055
1013304257027720009836164819203307201990417968010059033600021063376857613190905728010897286400475956135205
11139662700515592023973364864449665289863773920081714080448033289341845765608387040256278098748928017435658240012815133759385
12146889232904332053362108818911116224401551166016495992585088033563143950336112134280946688154207437576192613560813465602964061900800369605936607805
1315461233961511109611033502484986113932814038403258882423438181964824683995553817613875337510594563854638708816896437916433347993614141539328716805335311421440011361372997850695
141630166530242424002146015872120475555200434564754624010004175941760014343246647009281224849179473920057968455000135680135436880433254400128409919603789824339930475031347201013709170073600370609338222772825
Technical details
Node id
n-d3ab4b2119e206ba · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9275643d4265da83

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: 1,   6;
  2: 1,  18,    42;
  3: 1,  36,   276,     336;
  4: 1,  60,   960,    4200,     3024;
  5: 1,  90,  2460,   23400,    66024,     30240;
  6: 1, 126,  5250,   87360,   557424,   1086624,     332640;
  7: 1, 168,  9912,  255360,  2916144,  13349952,   18805248,    3991680;
  8: 1, 216, 17136,  632016, 11343024,  95001984,  325854144,  342486144,   51891840;
  9: 1, 270, 27720, 1386000, 35991648, 480664800, 3080911680, 8162985600, 6559128576, 726485760;
%F Formula
The row polynomials satisfy P(n,x) = (1+6*n*x)*P(n-1,x) - 5*x^2*d/dx 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[(1 + 6*n*x)*P[n - 1, x] - 5*x^2*D[P[n - 1, x], 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 + 6*n*x)*v[i - 1] - 5*x^2*deriv(v[i - 1], 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.