Real-rooted atlas

A130534 → D → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-8 + 12\,x - 6\,x^{2} + x^{3} + 12\,n - 12\,n x + 3\,n x^{2} - 6\,n^{2} + 3\,n^{2} x + n^{3}\right)\,P_{n-3}(x) + \left(-7 + 9\,x - 3\,x^{2} + 9\,n - 6\,n x - 3\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 3\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
222
312618
4706012142
5450510180201160
63248441021004203010208
726264406142352064408404297720
823624840370426938890720163801512561018008
923454004342080323190012654602835003654025207211507472
1025507152504570004100316018041100473319076230073920396090140581872
113018359526315484565519487602667907007912674015011766182952013860059401101848236544
1238631191048484089328788648203241214030001349706930290918628419699284015440244530858013226041957632
135319343411212188251857611949244507266722375720237282945905671531866933500568106162056819819041041012012156391738883328
1478431359564818659355624001916659268160113326733520043227968784011262105855020631050440268648380024747723015765750660660163801826268657962240
Technical details
Node id
n-7d59072621b39c7b · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-1db1b1b28f935ec6

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (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:       0;
  1:       0;
  2:       2;
  3:      12,       6;
  4:      70,      60,      12;
  5:     450,     510,     180,      20;
  6:    3248,    4410,    2100,     420,     30;
  7:   26264,   40614,   23520,    6440,    840,    42;
  8:  236248,  403704,  269388,   90720,  16380,  1512,   56;
  9: 2345400, 4342080, 3231900, 1265460, 283500, 36540, 2520, 72;
%F Formula
The row polynomials satisfy P(n,x) = (n^3-6*n^2+12*n-8+(3*n^2-12*n+12)*x+(3*n-6)*x^2+x^3)*P(n-3,x) - (3*n^2-9*n+7+(6*n-9)*x+3*x^2)*P(n-2,x) + (3*n-3+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*P[n - 2, x] + (3*n - 3 + 3*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] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*v[i - 3] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*v[i - 2] + (3*n - 3 + 3*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.