Real-rooted atlas

A075499 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 8\,x + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 12\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01121629
112411264201
21404009602561657
3160104057607936102415821
418422402240077056645124096170389
51112425667200435456989184520192163842032785
611447392169344177945679564801239040041779206553626546673
711801200037632058437124354560013969408015286272033488896262144376085653

All 15 rows

Technical details
Node id
n-9ada29ffbad64b45 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-6d39ca603c4714ce

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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,  12,    16;
  1: 1,  24,   112,      64;
  2: 1,  40,   400,     960,      256;
  3: 1,  60,  1040,    5760,     7936,      1024;
  4: 1,  84,  2240,   22400,    77056,     64512,       4096;
  5: 1, 112,  4256,   67200,   435456,    989184,     520192,       16384;
  6: 1, 144,  7392,  169344,  1779456,   7956480,   12390400,     4177920,       65536;
  7: 1, 180, 12000,  376320,  5843712,  43545600,  139694080,   152862720,    33488896,      262144;
  8: 1, 220, 18480,  760320, 16380672, 183794688, 1010606080,  2387968000,  1867841536,   268173312,    1048576;
  9: 1, 264, 27280, 1425600, 40710912, 642453504, 5421678592, 22600089600, 40075329536, 22682271744, 2146435072, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = (1+(4*n+8)*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 12*x + 16*x^2.
%t Mathematica
P[0, x_] := 1 + 12*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n + 8)*x)*P[n - 1, x] - 4*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 + 12*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (4*n + 8)*x)*v[i - 1] - 4*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.