Real-rooted atlas

A102365 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(x - x^{3} + n x^{3}\right)\,P_{n}(x) = \left(-2 - 7\,x - 4\,x^{2} - 2\,x^{3} + 2\,n + 5\,n x + 4\,n x^{2} - 4\,n x^{3} - 4\,n x^{4} - 2\,n x^{5} - n^{2} x + 4\,n^{2} x^{3} + 6\,n^{2} x^{4} + 3\,n^{2} x^{5} - n^{3} x^{3} - 2\,n^{3} x^{4} - n^{3} x^{5}\right)\,P_{n-2}(x) + \left(1 + x - x^{3} - x^{4} - n x + 2\,n x^{3} + n x^{4} - n x^{5} - n x^{6} - n^{2} x^{3} - n^{2} x^{4} + n^{2} x^{5} + n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(-2 - x - 2\,x^{2} + 2\,x^{4} + 2\,n x - 2\,n x^{3} - 4\,n x^{4} + 2\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(x - 2\,x^{3} + x^{5} + n x^{3} - n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
2224
3714324
435102544195
522686273816451995
61787841699693952440624570
71671793648141795822401792510987353430
818056011722822162232166058055917573350261885810805
9221118116322474355053543403243615716825335765428033660569107432325
103027304725036647062749620972200150442640174512931921818509505102177613716102205403200
1145818675241961537221190690902815999795024115467624304587795702964225038959519523599451305901149759409700
127596317885763098319122418247529703718300884003176749148251570052244604427297607666868768804752249751236535067430121223688641175
13136907048461149677244805652394517140589074680916240896705082247553184366915481898715142968393894898704438917664602273473550416823661472761332572116752175
14266508490248231500333151514120712260123360232585387265464261132375849100181067083299666786333607105622101636895376032916904324472760507073001058832177713847702431928014932890722513750
Technical details
Node id
n-2e1eaad3de0529b3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-0375a4d89aafc3ff

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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;
  1:       1;
  2:       2,        2;
  3:       7,       14,        3;
  4:      35,      102,       54,        4;
  5:     226,      862,      738,      164,        5;
  6:    1787,     8416,     9969,     3952,      440,       6;
  7:   16717,    93648,   141795,    82240,    17925,    1098,      7;
  8:  180560,  1172282,  2162232,  1660580,   559175,   73350,   2618,    8;
  9: 2211181, 16322474, 35505354, 34032436, 15716825, 3357654, 280336, 6056, 9;
%F Formula
The row polynomials satisfy (x+(n-1)*x^3)*P(n,x) = (2*n-2-(n^2-5*n+7)*x+(4*n-4)*x^2-(n^3-4*n^2+4*n+2)*x^3-(2*n^3-6*n^2+4*n)*x^4-(n^3-3*n^2+2*n)*x^5)*P(n-2,x) + (1-(n-1)*x-(n^2-2*n+1)*x^3-(n^2-n+1)*x^4+(n^2-n)*x^5+(n^2-n)*x^6)*d/dx P(n-2,x) - (2-(2*n-1)*x+2*x^2-(2*n^2-2*n)*x^3-(2*n^2-4*n+2)*x^4)*P(n-1,x) + (x+(n-2)*x^3-(n-1)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 - (n^2 - 5*n + 7)*x + (4*n - 4)*x^2 - (n^3 - 4*n^2 + 4*n + 2)*x^3 - (2*n^3 - 6*n^2 + 4*n)*x^4 - (n^3 - 3*n^2 + 2*n)*x^5)*P[n - 2, x] + (1 - (n - 1)*x - (n^2 - 2*n + 1)*x^3 - (n^2 - n + 1)*x^4 + (n^2 - n)*x^5 + (n^2 - n)*x^6)*D[P[n - 2, x], x] - (2 - (2*n - 1)*x + 2*x^2 - (2*n^2 - 2*n)*x^3 - (2*n^2 - 4*n + 2)*x^4)*P[n - 1, x] + (x + (n - 2)*x^3 - (n - 1)*x^5)*D[P[n - 1, x], x])/(x + (n - 1)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 - (n^2 - 5*n + 7)*x + (4*n - 4)*x^2 - (n^3 - 4*n^2 + 4*n + 2)*x^3 - (2*n^3 - 6*n^2 + 4*n)*x^4 - (n^3 - 3*n^2 + 2*n)*x^5)*v[i - 2] + (1 - (n - 1)*x - (n^2 - 2*n + 1)*x^3 - (n^2 - n + 1)*x^4 + (n^2 - n)*x^5 + (n^2 - n)*x^6)*deriv(v[i - 2], x) - (2 - (2*n - 1)*x + 2*x^2 - (2*n^2 - 2*n)*x^3 - (2*n^2 - 4*n + 2)*x^4)*v[i - 1] + (x + (n - 2)*x^3 - (n - 1)*x^5)*deriv(v[i - 1], x))/(x + (n - 1)*x^3)); 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

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.