Real-rooted atlas

A143491 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 6\,x - x^{2} - 6\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(7 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 19 + 9\,x + x^{2}\)
\(P_{1}(x) = 107 + 71\,x + 15\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0199129
110771151194
27025801732211478
352745104191034530112664
44471248860215594915610391120696
542256850900425504568929108509944911267440
644071205753736319339298470418456921532152660114546640
7502927207029093642461556145627643126732435981394802238721181212480

All 15 rows

Technical details
Node id
n-b291e351cce4e0ed · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7195245dea7f3067

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:         19,           9,          1;
  1:        107,          71,         15,          1;
  2:        702,         580,        173,         22,         1;
  3:       5274,        5104,       1910,        345,        30,         1;
  4:      44712,       48860,      21559,       4915,       610,        39,        1;
  5:     422568,      509004,     255045,      68929,     10850,       994,       49,       1;
  6:    4407120,     5753736,    3193392,     984704,    184569,     21532,     1526,      60,      1;
  7:   50292720,    70290936,   42461556,   14562764,   3126732,    435981,    39480,    2238,     72,    1;
  8:  623471040,   924118272,  599807916,  224709140,  53850635,   8630433,   937083,   68070,   3165,   85,  1;
  9: 8344624320, 13020978816, 8991695196, 3630938036, 953466855, 171076499, 21412545, 1870143, 111705, 4345, 99, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(2*n+6)*x+x^2)*P(n-2,x) + (2*n+7+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 19 + 9*x + x^2, P(1,x) = 107 + 71*x + 15*x^2 + x^3.
%t Mathematica
P[0, x_] := 19 + 9*x + x^2;
P[1, x_] := 107 + 71*x + 15*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (2*n + 6)*x + x^2)*P[n - 2, x] + (2*n + 7 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 19 + 9*x + x^2; v[2] = 107 + 71*x + 15*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (2*n + 6)*x + x^2)*v[i - 2] + (2*n + 7 + 2*x)*v[i - 1]); 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.