Real-rooted atlas

A156992 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,x + 6\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 24\,x^{2} + 16\,x^{3} + 2\,n + 24\,n x + 64\,n x^{2} + 48\,n x^{3} + 12\,n^{2} x + 48\,n^{2} x^{2} + 48\,n^{2} x^{3} + 8\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
143236
2243848641272
31924608207362457650112
41920614404147209830407680002229120
5230409216008294400294912004608000026542080111352320
6322560154828801741824008257536001935360000222953472010115481606192184320
751609602890137603901685760231211008007225344000012485394432011329339392042278584320379996323840

All 15 rows

Technical details
Node id
n-ba8807b37d39c329 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

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,           32;
  2:         24,          384,           864;
  3:        192,         4608,         20736,          24576;
  4:       1920,        61440,        414720,         983040,         768000;
  5:      23040,       921600,       8294400,       29491200,       46080000,        26542080;
  6:     322560,     15482880,     174182400,      825753600,     1935360000,      2229534720,      1011548160;
  7:    5160960,    289013760,    3901685760,    23121100800,    72253440000,    124853944320,    113293393920,     42278584320;
  8:   92897280,   5945425920,   93640458240,   665887703040,  2601123840000,   5992989327360,   8157124362240,   6088116142080,   1926317998080;
  9: 1857945600, 133772083200, 2407897497600, 19976631091200, 93640458240000, 269684519731200, 489427461734400, 547930452787200, 346737239654400, 95126814720000;
%F Formula
The row polynomials satisfy (1+(6*n-2)*x+4*n^2*x^2)*P(n,x) = (2*n+2+(12*n^2+24*n+12)*x+(8*n^3+48*n^2+64*n+24)*x^2+(16*n^3+48*n^2+48*n+16)*x^3)*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[((2*n + 2 + (12*n^2 + 24*n + 12)*x + (8*n^3 + 48*n^2 + 64*n + 24)*x^2 + (16*n^3 + 48*n^2 + 48*n + 16)*x^3)*P[n - 1, x])/(1 + (6*n - 2)*x + 4*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] = ((2*n + 2 + (12*n^2 + 24*n + 12)*x + (8*n^3 + 48*n^2 + 64*n + 24)*x^2 + (16*n^3 + 48*n^2 + 48*n + 16)*x^3)*v[i - 1])/(1 + (6*n - 2)*x + 4*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

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.