Real-rooted atlas

A142459 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-7 + 4\,n\right)\,P_{n}(x) = \left(-9 + 198\,x + 231\,x^{2} + 12\,n - 336\,n x - 524\,n x^{2} + 96\,n^{2} x + 336\,n^{2} x^{2} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-36\,x - 48\,x^{2} + 84\,x^{3} + 48\,n x + 112\,n x^{2} - 160\,n x^{3} - 64\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 98\,x - 8\,n - 112\,n x + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-28\,x + 28\,x^{2} + 16\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
3120324
411181774300
516163186123255040
613114406865424877856105840
711561244558114104487426354683672661120
8178110450200728435660355445759004014273385877837760

All 15 rows

Technical details
Node id
n-c3dfc3a32f923c60 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b99dd02e1a9529cf

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,       2;
   3: 1,      20,         3;
   4: 1,     118,       177,          4;
   5: 1,     616,      3186,       1232,           5;
   6: 1,    3114,     40686,      54248,        7785,           6;
   7: 1,   15612,    445581,    1410448,      742635,       46836,           7;
   8: 1,   78110,   4502007,   28435660,    35544575,     9004014,      273385,          8;
   9: 1,  390608,  43447188,  495719776,  1208515550,   743579664,   101376772,    1562432,       9;
  10: 1, 1953106, 408016140, 7892596688, 33557408470, 40268890164, 13812044204, 1088043040, 8788977, 10;
%F Formula
The row polynomials satisfy (4*n-7)*P(n,x) = (12*n-9+(96*n^2-336*n+198)*x-(64*n^3-336*n^2+524*n-231)*x^2)*P(n-2,x) + ((48*n-36)*x-(64*n^2-112*n+48)*x^2+(64*n^2-160*n+84)*x^3)*d/dx P(n-2,x) - (8*n-2-(32*n^2-112*n+98)*x)*P(n-1,x) + ((16*n-28)*x-(16*n-28)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n - 9 + (96*n^2 - 336*n + 198)*x - (64*n^3 - 336*n^2 + 524*n - 231)*x^2)*P[n - 2, x] + ((48*n - 36)*x - (64*n^2 - 112*n + 48)*x^2 + (64*n^2 - 160*n + 84)*x^3)*D[P[n - 2, x], x] - (8*n - 2 - (32*n^2 - 112*n + 98)*x)*P[n - 1, x] + ((16*n - 28)*x - (16*n - 28)*x^2)*D[P[n - 1, x], x])/(4*n - 7)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = ((12*n - 9 + (96*n^2 - 336*n + 198)*x - (64*n^3 - 336*n^2 + 524*n - 231)*x^2)*v[i - 2] + ((48*n - 36)*x - (64*n^2 - 112*n + 48)*x^2 + (64*n^2 - 160*n + 84)*x^3)*deriv(v[i - 2], x) - (8*n - 2 - (32*n^2 - 112*n + 98)*x)*v[i - 1] + ((16*n - 28)*x - (16*n - 28)*x^2)*deriv(v[i - 1], x))/(4*n - 7)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.