Real-rooted atlas

A103327 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-6\,n x - 16\,n^{2} + 84\,n^{2} x - 9\,n^{2} x^{2} + 32\,n^{3} - 216\,n^{3} x + 54\,n^{3} x^{2} + 144\,n^{4} x - 108\,n^{4} x^{2} + 72\,n^{5} x^{2}\right)\,P_{n}(x) = \left(108\,x^{2} - 81\,x^{3} - 24\,n x + 432\,n x^{2} + 378\,n x^{3} - 64\,n^{2} - 624\,n^{2} x - 1368\,n^{2} x^{2} - 351\,n^{2} x^{3} + 128\,n^{3} + 288\,n^{3} x - 1080\,n^{3} x^{2} - 1026\,n^{3} x^{3} + 576\,n^{4} x + 1728\,n^{4} x^{2} + 540\,n^{4} x^{3} + 288\,n^{5} x^{2} + 648\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(12\,x^{2} + 18\,x^{3} - 54\,x^{4} + 72\,n x^{2} + 288\,n x^{3} + 216\,n x^{4} - 64\,n^{2} x - 336\,n^{2} x^{2} - 468\,n^{2} x^{3} - 108\,n^{2} x^{4} - 288\,n^{3} x^{2} - 1152\,n^{3} x^{3} - 864\,n^{3} x^{4} - 144\,n^{4} x^{3} - 432\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141216
21614481241
36496019444323400
425653761944017280202544377
51024276481451522419201215008748545992
6409613516893312023224322268000734832357216433369
7163846389765474304182476802721600017635968400075213996873370032

All 15 rows

Technical details
Node id
n-f1b9721ef097c1eb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9637c02c4490662e

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,       12;
  2:     16,      144,        81;
  3:     64,      960,      1944,       432;
  4:    256,     5376,     19440,     17280,       2025;
  5:   1024,    27648,    145152,    241920,     121500,       8748;
  6:   4096,   135168,    933120,   2322432,    2268000,     734832,      35721;
  7:  16384,   638976,   5474304,  18247680,   27216000,   17635968,    4000752,    139968;
  8:  65536,  2949120,  30191616, 126517248,  256608000,  258660864,  120022560,  20155392,   531441;
  9: 262144, 13369344, 159252480, 805109760, 2075673600, 2882221056, 2112397056, 739031040, 95659380, 1968300;
%F Formula
The row polynomials satisfy (32*n^3-16*n^2+(144*n^4-216*n^3+84*n^2-6*n)*x+(72*n^5-108*n^4+54*n^3-9*n^2)*x^2)*P(n,x) = (128*n^3-64*n^2+(576*n^4+288*n^3-624*n^2-24*n)*x+(288*n^5+1728*n^4-1080*n^3-1368*n^2+432*n+108)*x^2+(648*n^5+540*n^4-1026*n^3-351*n^2+378*n-81)*x^3)*P(n-1,x) - (64*n^2*x+(288*n^3+336*n^2-72*n-12)*x^2+(144*n^4+1152*n^3+468*n^2-288*n-18)*x^3+(432*n^4+864*n^3+108*n^2-216*n+54)*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[((128*n^3 - 64*n^2 + (576*n^4 + 288*n^3 - 624*n^2 - 24*n)*x + (288*n^5 + 1728*n^4 - 1080*n^3 - 1368*n^2 + 432*n + 108)*x^2 + (648*n^5 + 540*n^4 - 1026*n^3 - 351*n^2 + 378*n - 81)*x^3)*P[n - 1, x] - (64*n^2*x + (288*n^3 + 336*n^2 - 72*n - 12)*x^2 + (144*n^4 + 1152*n^3 + 468*n^2 - 288*n - 18)*x^3 + (432*n^4 + 864*n^3 + 108*n^2 - 216*n + 54)*x^4)*D[P[n - 1, x], x])/(32*n^3 - 16*n^2 + (144*n^4 - 216*n^3 + 84*n^2 - 6*n)*x + (72*n^5 - 108*n^4 + 54*n^3 - 9*n^2)*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] = ((128*n^3 - 64*n^2 + (576*n^4 + 288*n^3 - 624*n^2 - 24*n)*x + (288*n^5 + 1728*n^4 - 1080*n^3 - 1368*n^2 + 432*n + 108)*x^2 + (648*n^5 + 540*n^4 - 1026*n^3 - 351*n^2 + 378*n - 81)*x^3)*v[i - 1] - (64*n^2*x + (288*n^3 + 336*n^2 - 72*n - 12)*x^2 + (144*n^4 + 1152*n^3 + 468*n^2 - 288*n - 18)*x^3 + (432*n^4 + 864*n^3 + 108*n^2 - 216*n + 54)*x^4)*deriv(v[i - 1], x))/(32*n^3 - 16*n^2 + (144*n^4 - 216*n^3 + 84*n^2 - 6*n)*x + (72*n^5 - 108*n^4 + 54*n^3 - 9*n^2)*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

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