Real-rooted atlas

A156992 → basis A008277 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2\,n - 8\,n x + 9\,n x^{2} - 2\,n^{2} - 8\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,n x + 8\,n x^{2} + 4\,n x^{3} - 4\,n^{2} x + 8\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-n x^{2} + 4\,n x^{3} - n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(3 + 4\,x + 3\,n + 4\,n x\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2} + 2\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
22412054198
3192182419443844344
419203120059400268803000122400
52304060768018468001440000360000259204303440
6322560134064006128136073382400308700004898880246960184408560
751609603320755202202318720380685312023990400006299596806914880025804809447137280
8928972809151833600860080032002065454899201830820320007077901824012695719680102187008029393280569406257280
9185794560027827089920036462363840001185630566400014212195200000759091253760019790386560002577899520001587237120036288000039838842489600
10408748032009264689280000167321605651200723400989696000114178513680000080962143736320028780823111040053954740224000533488032000025945920000048299328003198796874380800
11980995276800335490804633600828158368181760046973586968985600957722984064000008773824790241280040853406996864000103380654440448001459623255552000113283878400000446285790720068976230400291871100168524800
1225505877196800131351531351040004404554574900480003244679753630515200842439026100672000097770379852873728005801233880835993600190322783289999360035984450623080960039625646860800000247816124075520080702189568000105236651520030006215899051392000
1371416456151040055310476079738880025079262943198464000238090029696110592000778626151535232000000112798543022005800960083604508515105177600034588641219497164800084101049036510105600123218842921574400001087589546268825600559768320663552001532245646131200170869450752003449834239705575321600
142142493684531200024932564109573120000152342306226881971200018525532572360253440000756565887091734720000001352567950462671022080001233551127517857308160006307432445744381952000019181594711990430720000359053774221542400000041940831823104614400030330303905341440000130719706685568000030500196959232000294226732800000440639939656178406144000
Technical details
Node id
n-2bc5bb45e09155dc · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-fefd2b9ccc27c8c7

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:         24,          120,            54;
  3:        192,         1824,          1944,            384;
  4:       1920,        31200,         59400,          26880,           3000;
  5:      23040,       607680,       1846800,        1440000,         360000,         25920;
  6:     322560,     13406400,      61281360,       73382400,       30870000,       4898880,        246960;
  7:    5160960,    332075520,    2202318720,     3806853120,     2399040000,     629959680,      69148800,      2580480;
  8:   92897280,   9151833600,   86008003200,   206545489920,   183082032000,   70779018240,   12695719680,   1021870080,    29393280;
  9: 1857945600, 278270899200, 3646236384000, 11856305664000, 14212195200000, 7590912537600, 1979038656000, 257789952000, 15872371200, 362880000;
%F Formula
The row polynomials satisfy P(n,x) = -(2*n^2+2*n+(8*n^2+8*n)*x-(9*n^2+9*n)*x^2)*P(n-2,x) - ((4*n^2+4*n)*x-(8*n^2+8*n)*x^2-(4*n^2+4*n)*x^3)*d/dx P(n-2,x) - ((n^2+n)*x^2-(4*n^2+4*n)*x^3)*d^2/dx^2 P(n-2,x) + (3*n+3+(4*n+4)*x)*P(n-1,x) + ((2*n+2)*x-(4*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 + 2*n + (8*n^2 + 8*n)*x - (9*n^2 + 9*n)*x^2)*P[n - 2, x] - ((4*n^2 + 4*n)*x - (8*n^2 + 8*n)*x^2 - (4*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] - ((n^2 + n)*x^2 - (4*n^2 + 4*n)*x^3)*D[P[n - 2, x], {x, 2}] + (3*n + 3 + (4*n + 4)*x)*P[n - 1, x] + ((2*n + 2)*x - (4*n + 4)*x^2)*D[P[n - 1, x], x]];
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] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(2*n^2 + 2*n + (8*n^2 + 8*n)*x - (9*n^2 + 9*n)*x^2)*v[i - 2] - ((4*n^2 + 4*n)*x - (8*n^2 + 8*n)*x^2 - (4*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) - ((n^2 + n)*x^2 - (4*n^2 + 4*n)*x^3)*deriv(deriv(v[i - 2], x), x) + (3*n + 3 + (4*n + 4)*x)*v[i - 1] + ((2*n + 2)*x - (4*n + 4)*x^2)*deriv(v[i - 1], x)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.