Real-rooted atlas

A132812 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(8\,n - 4\,n x + 12\,n^{2} + 4\,n^{2} x + 4\,n^{3} + 15\,n^{3} x + 2\,n^{3} x^{2} + 6\,n^{4} x + 3\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(8 - 4\,x + 32\,n + 42\,n x + 2\,n x^{3} + 40\,n^{2} + 118\,n^{2} x + 58\,n^{2} x^{2} + 9\,n^{2} x^{3} + 16\,n^{3} + 96\,n^{3} x + 90\,n^{3} x^{2} + 20\,n^{3} x^{3} + 24\,n^{4} x + 40\,n^{4} x^{2} + 14\,n^{4} x^{3} + 4\,n^{5} x^{2} + 3\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-8\,x - 14\,x^{2} - 8\,x^{3} - 2\,x^{4} - 16\,n x - 47\,n x^{2} - 40\,n x^{3} - 9\,n x^{4} - 8\,n^{2} x - 44\,n^{2} x^{2} - 52\,n^{2} x^{3} - 16\,n^{2} x^{4} - 12\,n^{3} x^{2} - 22\,n^{3} x^{3} - 10\,n^{3} x^{4} - 2\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
2156027102
35633632464780
42101680252011201255655
579279201620011520300021639648
63003360369355592400393756804343271516
711440160160504504640640385000110880137205121826856
84375870012825945924036032315315012972962716562534472912122685
9167960302328012888720237619202293200012383280374556059904043740100079546500
1064664612932920623551501330243201531530001029188164120116096096001216215715001331517130658
1124961445491516829546748071628480095931000077358758438792476812063744022453200231000011180417283335500116
129657700231784800137680171237394614405715888750538748496032590958401277033472318995820486200004152720168480219721369147891
13374421609734961606327725040190372582403272028760035337910608250950379681191897907237712394727759752009883473671285762460642744136101563640
141454226754071834900287485584759491587560018126295687522086194130018037058539510096545888038890907055101846745001760493735190466640117649353498603375862381294200
Technical details
Node id
n-88d2e25bf44a0a97 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-097fa7332775e3f8

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 0)
  • 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:      4,       8;
  2:     15,      60,       27;
  3:     56,     336,      324,       64;
  4:    210,    1680,     2520,     1120,      125;
  5:    792,    7920,    16200,    11520,     3000,      216;
  6:   3003,   36036,    93555,    92400,    39375,     6804,     343;
  7:  11440,  160160,   504504,   640640,   385000,   110880,   13720,    512;
  8:  43758,  700128,  2594592,  4036032,  3153150,  1297296,  271656,  25344,   729;
  9: 167960, 3023280, 12888720, 23761920, 22932000, 12383280, 3745560, 599040, 43740, 1000;
%F Formula
The row polynomials satisfy (4*n^3+12*n^2+8*n+(6*n^4+15*n^3+4*n^2-4*n)*x+(n^5+3*n^4+2*n^3)*x^2)*P(n,x) = (16*n^3+40*n^2+32*n+8+(24*n^4+96*n^3+118*n^2+42*n-4)*x+(4*n^5+40*n^4+90*n^3+58*n^2)*x^2+(3*n^5+14*n^4+20*n^3+9*n^2+2*n)*x^3)*P(n-1,x) - ((8*n^2+16*n+8)*x+(12*n^3+44*n^2+47*n+14)*x^2+(2*n^4+22*n^3+52*n^2+40*n+8)*x^3+(2*n^4+10*n^3+16*n^2+9*n+2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 + 40*n^2 + 32*n + 8 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (4*n^5 + 40*n^4 + 90*n^3 + 58*n^2)*x^2 + (3*n^5 + 14*n^4 + 20*n^3 + 9*n^2 + 2*n)*x^3)*P[n - 1, x] - ((8*n^2 + 16*n + 8)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (2*n^4 + 22*n^3 + 52*n^2 + 40*n + 8)*x^3 + (2*n^4 + 10*n^3 + 16*n^2 + 9*n + 2)*x^4)*D[P[n - 1, x], x])/(4*n^3 + 12*n^2 + 8*n + (6*n^4 + 15*n^3 + 4*n^2 - 4*n)*x + (n^5 + 3*n^4 + 2*n^3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 40*n^2 + 32*n + 8 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (4*n^5 + 40*n^4 + 90*n^3 + 58*n^2)*x^2 + (3*n^5 + 14*n^4 + 20*n^3 + 9*n^2 + 2*n)*x^3)*v[i - 1] - ((8*n^2 + 16*n + 8)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (2*n^4 + 22*n^3 + 52*n^2 + 40*n + 8)*x^3 + (2*n^4 + 10*n^3 + 16*n^2 + 9*n + 2)*x^4)*deriv(v[i - 1], x))/(4*n^3 + 12*n^2 + 8*n + (6*n^4 + 15*n^3 + 4*n^2 - 4*n)*x + (n^5 + 3*n^4 + 2*n^3)*x^2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023