Real-rooted atlas

A143491 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 6\,x - x^{2} - 6\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(7 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 7 + 2\,x\)
\(P_{1}(x) = 47 + 24\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0729
14724374
2342238544638
32754235073510055944
4245522430894051780165660216
52411282678761211072856037102527662640
625927203161016161288444707672800697236487893840
730334320399566162248324870683481359645164052121385049101378880

All 15 rows

Technical details
Node id
n-632105466d3dab2d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-771b79ba348cbb4e

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          7,          2;
  1:         47,         24,          3;
  2:        342,        238,         54,          4;
  3:       2754,       2350,        735,        100,         5;
  4:      24552,      24308,       9405,       1780,       165,        6;
  5:     241128,     267876,     121107,      28560,      3710,      252,       7;
  6:    2592720,    3161016,    1612884,     447076,     72800,     6972,     364,      8;
  7:   30334320,   39956616,   22483248,    7068348,   1359645,   164052,   12138,    504,     9;
  8:  383970240,  540148032,  329733900,  114797840,  25151175,  3600198,  337050,  19920,   675,  10;
  9: 5231113920, 7789864896, 5096762748, 1932017120, 470462575, 76983984, 8581881, 644160, 31185, 880, 11;
%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) = 7 + 2*x, P(1,x) = 47 + 24*x + 3*x^2.
%t Mathematica
P[0, x_] := 7 + 2*x;
P[1, x_] := 47 + 24*x + 3*x^2;
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] = 7 + 2*x; v[2] = 47 + 24*x + 3*x^2; 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.