Real-rooted atlas

A345013 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
26201541
3106010556231
4151404205042101289
5212801260252023107927183
628504315092401386010296300340081
73684069302772060060720724504511440224143
84513201386072072210210360360360360194480437581256465
9551980257401681686306301441440204204017503208314021679607059735
1066286045045360360168168049008969189180110853608314020352716064664639753273
117840047507572072040840801470268834918884554268005819814038798760148728582496144224298231
1291546012012013613609189180399072961163962802327925603200897702974571601784742966240360096577001267854873
13105728018564024504481939938099768240349188840853572720147241294217847429601487285800811246800260757900374421607178461215
141209520278460423259238798760232792560960269310280459608058896517688923714800966735770073012212003650610600108582264014542267540704778785
Technical details
Node id
n-136b2136a8d1f9f3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-3961bf8dfc4504b6

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:  6,   20,    15;
  3: 10,   60,   105,     56;
  4: 15,  140,   420,    504,    210;
  5: 21,  280,  1260,   2520,   2310,     792;
  6: 28,  504,  3150,   9240,  13860,   10296,    3003;
  7: 36,  840,  6930,  27720,  60060,   72072,   45045,   11440;
  8: 45, 1320, 13860,  72072, 210210,  360360,  360360,  194480,  43758;
  9: 55, 1980, 25740, 168168, 630630, 1441440, 2042040, 1750320, 831402, 167960;
%F Formula
The row polynomials satisfy (n^2+2*n)*P(n,x) = (n^2+4*n+4+(2*n^2+6*n+4)*x)*P(n-1,x) + ((2*n+2)*x+(2*n+2)*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[Together[((n^2 + 4*n + 4 + (2*n^2 + 6*n + 4)*x)*P[n - 1, x] + ((2*n + 2)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n)]];
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^2 + 4*n + 4 + (2*n^2 + 6*n + 4)*x)*v[i - 1] + ((2*n + 2)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n)); 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

Candidate equalities ?

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.