Real-rooted atlas

A367270 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-n + 4\,n x + 5\,n^{2} - 28\,n^{2} x - 8\,n^{3} + 72\,n^{3} x - 16\,n^{3} x^{2} + 4\,n^{4} - 72\,n^{4} x + 48\,n^{4} x^{2} + 24\,n^{5} x - 48\,n^{5} x^{2} + 16\,n^{6} x^{2}\right)\,P_{n}(x) = \left(2 + 24\,x + 96\,x^{2} + 128\,x^{3} - 14\,n - 136\,n x - 432\,n x^{2} - 320\,n x^{3} + 36\,n^{2} + 240\,n^{2} x + 352\,n^{2} x^{2} - 40\,n^{3} - 80\,n^{3} x + 544\,n^{3} x^{2} + 576\,n^{3} x^{3} + 16\,n^{4} - 144\,n^{4} x - 880\,n^{4} x^{2} - 384\,n^{4} x^{3} + 96\,n^{5} x + 256\,n^{5} x^{2} - 192\,n^{5} x^{3} + 64\,n^{6} x^{2} + 192\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(x + 12\,x^{2} + 48\,x^{3} + 64\,x^{4} - 6\,n x - 56\,n x^{2} - 160\,n x^{3} - 128\,n x^{4} + 12\,n^{2} x + 64\,n^{2} x^{2} + 48\,n^{2} x^{3} - 64\,n^{2} x^{4} - 8\,n^{3} x + 16\,n^{3} x^{2} + 256\,n^{3} x^{3} + 256\,n^{3} x^{4} - 48\,n^{4} x^{2} - 208\,n^{4} x^{3} - 64\,n^{4} x^{4} - 32\,n^{5} x^{3} - 128\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
233235
310192432634
435960432040969411
51264480302406144032000128286
6462201601814405734406720002211841668686
7171688704997920430080080640006193152140492821051220
864353843845189184283852807392000092897280505774088388608259748579
9243101647360259459201722040325765760001021870080927252480377487360477757443150783286
1092378700128012602304098402304040360320009299017728120542822408304721920262766592026214400037701003546
11352716295609605986094405376983040261381120007439214182412656996352012595494912068319313920173015040001395654656446077145196
12135207812415603227935107202837852160015962918400054199989043211475676692481511459389440119558799360052481228800010886106316872477573125228462775630
135200300519197952128501493121456764108809311702400003677856399360934447959244815416885772288162599967129601049624576000038101372108806595459153923685954355260792228105324
14200583002163324800584097696007310307164165235246016000235995785625607008359694336013948610936832018536396252774416059256012800086363110113280263818366156803870252072960184146722816701952022939836
Technical details
Node id
n-539045161dc20702 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-572eea917081eb98

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:     3,      32;
  3:    10,     192,      432;
  4:    35,     960,     4320,      4096;
  5:   126,    4480,    30240,     61440,     32000;
  6:   462,   20160,   181440,    573440,    672000,     221184;
  7:  1716,   88704,   997920,   4300800,   8064000,    6193152,   1404928;
  8:  6435,  384384,  5189184,  28385280,  73920000,   92897280,  50577408,   8388608;
  9: 24310, 1647360, 25945920, 172204032, 576576000, 1021870080, 927252480, 377487360, 47775744;
%F Formula
The row polynomials satisfy (4*n^4-8*n^3+5*n^2-n+(24*n^5-72*n^4+72*n^3-28*n^2+4*n)*x+(16*n^6-48*n^5+48*n^4-16*n^3)*x^2)*P(n,x) = (16*n^4-40*n^3+36*n^2-14*n+2+(96*n^5-144*n^4-80*n^3+240*n^2-136*n+24)*x+(64*n^6+256*n^5-880*n^4+544*n^3+352*n^2-432*n+96)*x^2+(192*n^6-192*n^5-384*n^4+576*n^3-320*n+128)*x^3)*P(n-1,x) - ((8*n^3-12*n^2+6*n-1)*x+(48*n^4-16*n^3-64*n^2+56*n-12)*x^2+(32*n^5+208*n^4-256*n^3-48*n^2+160*n-48)*x^3+(128*n^5+64*n^4-256*n^3+64*n^2+128*n-64)*x^4)*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[((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (96*n^5 - 144*n^4 - 80*n^3 + 240*n^2 - 136*n + 24)*x + (64*n^6 + 256*n^5 - 880*n^4 + 544*n^3 + 352*n^2 - 432*n + 96)*x^2 + (192*n^6 - 192*n^5 - 384*n^4 + 576*n^3 - 320*n + 128)*x^3)*P[n - 1, x] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (48*n^4 - 16*n^3 - 64*n^2 + 56*n - 12)*x^2 + (32*n^5 + 208*n^4 - 256*n^3 - 48*n^2 + 160*n - 48)*x^3 + (128*n^5 + 64*n^4 - 256*n^3 + 64*n^2 + 128*n - 64)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 8*n^3 + 5*n^2 - n + (24*n^5 - 72*n^4 + 72*n^3 - 28*n^2 + 4*n)*x + (16*n^6 - 48*n^5 + 48*n^4 - 16*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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (96*n^5 - 144*n^4 - 80*n^3 + 240*n^2 - 136*n + 24)*x + (64*n^6 + 256*n^5 - 880*n^4 + 544*n^3 + 352*n^2 - 432*n + 96)*x^2 + (192*n^6 - 192*n^5 - 384*n^4 + 576*n^3 - 320*n + 128)*x^3)*v[i - 1] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (48*n^4 - 16*n^3 - 64*n^2 + 56*n - 12)*x^2 + (32*n^5 + 208*n^4 - 256*n^3 - 48*n^2 + 160*n - 48)*x^3 + (128*n^5 + 64*n^4 - 256*n^3 + 64*n^2 + 128*n - 64)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 8*n^3 + 5*n^2 - n + (24*n^5 - 72*n^4 + 72*n^3 - 28*n^2 + 4*n)*x + (16*n^6 - 48*n^5 + 48*n^4 - 16*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

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