Real-rooted atlas

A391045 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(54 - 142\,x + 70\,x^{2} - 8\,x^{3} - 300\,n + 684\,n x - 304\,n x^{2} + 32\,n x^{3} + 650\,n^{2} - 1198\,n^{2} x + 446\,n^{2} x^{2} - 40\,n^{2} x^{3} - 712\,n^{3} + 960\,n^{3} x - 260\,n^{3} x^{2} + 16\,n^{3} x^{3} + 420\,n^{4} - 352\,n^{4} x + 48\,n^{4} x^{2} - 128\,n^{5} + 48\,n^{5} x + 16\,n^{6}\right)\,P_{n}(x) = \left(18 + 690\,x - 2666\,x^{2} + 1604\,x^{3} - 208\,x^{4} - 108\,n - 3668\,n x + 10440\,n x^{2} - 5304\,n x^{3} + 608\,n x^{4} + 262\,n^{2} + 7558\,n^{2} x - 15114\,n^{2} x^{2} + 5876\,n^{2} x^{3} - 528\,n^{2} x^{4} - 336\,n^{3} - 7868\,n^{3} x + 10396\,n^{3} x^{2} - 2684\,n^{3} x^{3} + 144\,n^{3} x^{4} + 244\,n^{4} + 4400\,n^{4} x - 3432\,n^{4} x^{2} + 432\,n^{4} x^{3} - 96\,n^{5} - 1256\,n^{5} x + 432\,n^{5} x^{2} + 16\,n^{6} + 144\,n^{6} x\right)\,P_{n-1}(x) + \left(-78\,x + 890\,x^{2} - 2478\,x^{3} + 1706\,x^{4} - 248\,x^{5} + 374\,n x - 3834\,n x^{2} + 8264\,n x^{3} - 4434\,n x^{4} + 536\,n x^{5} - 660\,n^{2} x + 6052\,n^{2} x^{2} - 9080\,n^{2} x^{3} + 3044\,n^{2} x^{4} - 208\,n^{2} x^{5} + 548\,n^{3} x - 4496\,n^{3} x^{2} + 4048\,n^{3} x^{3} - 624\,n^{3} x^{4} - 216\,n^{4} x + 1584\,n^{4} x^{2} - 624\,n^{4} x^{3} + 32\,n^{5} x - 208\,n^{5} x^{2}\right)\,P_{n-1}'(x) + \left(-21\,x^{2} + 167\,x^{3} - 401\,x^{4} + 288\,x^{5} - 48\,x^{6} + 91\,n x^{2} - 615\,n x^{3} + 1155\,n x^{4} - 624\,n x^{5} + 80\,n x^{6} - 136\,n^{2} x^{2} + 766\,n^{2} x^{3} - 948\,n^{2} x^{4} + 240\,n^{2} x^{5} + 86\,n^{3} x^{2} - 404\,n^{3} x^{3} + 240\,n^{3} x^{4} - 20\,n^{4} x^{2} + 80\,n^{4} x^{3}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2325
3914326
41978424143
5332843661005788
6517702040126020564332
7731722802599603570378723735
899337424738539003801088206448129593
91296008640922205842718101219681965610329705288
10163995414601673500014332501117116343728403921575103827204
112011559030145520971206003480740703639339308751607771523101120714008
12243233425758505304420210711603826468832144112122694002052765141570327812111850840
132893368410330981218910064358415162698976201502224121472208346846504498780246246452413602722207
143394713817599922590416417579061559259400210242872649112165924101437349038458093098724116846097143241856087
1539364274287114151597000438127690190395605444002928975473950768363738375012609597002198936741834123266502880501517408111666
Technical details
Node id
n-6fe6856f0b35d23f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-135fd721cf11d97a

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   3,    2;
   3:   9,   14,      3;
   4:  19,   78,     42,      4;
   5:  33,  284,    366,    100,       5;
   6:  51,  770,   2040,   1260,     205,       6;
   7:  73, 1722,   8025,   9960,    3570,     378,      7;
   8:  99, 3374,  24738,  53900,   38010,    8820,    644,     8;
   9: 129, 6008,  64092, 220584,  271810,  121968,  19656,  1032,    9;
  10: 163, 9954, 146016, 735000, 1433250, 1117116, 343728, 40392, 1575, 10;
%F Formula
The row polynomials satisfy (16*n^6-128*n^5+420*n^4-712*n^3+650*n^2-300*n+54+(48*n^5-352*n^4+960*n^3-1198*n^2+684*n-142)*x+(48*n^4-260*n^3+446*n^2-304*n+70)*x^2+(16*n^3-40*n^2+32*n-8)*x^3)*P(n,x) = (16*n^6-96*n^5+244*n^4-336*n^3+262*n^2-108*n+18+(144*n^6-1256*n^5+4400*n^4-7868*n^3+7558*n^2-3668*n+690)*x+(432*n^5-3432*n^4+10396*n^3-15114*n^2+10440*n-2666)*x^2+(432*n^4-2684*n^3+5876*n^2-5304*n+1604)*x^3+(144*n^3-528*n^2+608*n-208)*x^4)*P(n-1,x) + ((32*n^5-216*n^4+548*n^3-660*n^2+374*n-78)*x-(208*n^5-1584*n^4+4496*n^3-6052*n^2+3834*n-890)*x^2-(624*n^4-4048*n^3+9080*n^2-8264*n+2478)*x^3-(624*n^3-3044*n^2+4434*n-1706)*x^4-(208*n^2-536*n+248)*x^5)*d/dx P(n-1,x) - ((20*n^4-86*n^3+136*n^2-91*n+21)*x^2-(80*n^4-404*n^3+766*n^2-615*n+167)*x^3-(240*n^3-948*n^2+1155*n-401)*x^4-(240*n^2-624*n+288)*x^5-(80*n-48)*x^6)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^6 - 96*n^5 + 244*n^4 - 336*n^3 + 262*n^2 - 108*n + 18 + (144*n^6 - 1256*n^5 + 4400*n^4 - 7868*n^3 + 7558*n^2 - 3668*n + 690)*x + (432*n^5 - 3432*n^4 + 10396*n^3 - 15114*n^2 + 10440*n - 2666)*x^2 + (432*n^4 - 2684*n^3 + 5876*n^2 - 5304*n + 1604)*x^3 + (144*n^3 - 528*n^2 + 608*n - 208)*x^4)*P[n - 1, x] + ((32*n^5 - 216*n^4 + 548*n^3 - 660*n^2 + 374*n - 78)*x - (208*n^5 - 1584*n^4 + 4496*n^3 - 6052*n^2 + 3834*n - 890)*x^2 - (624*n^4 - 4048*n^3 + 9080*n^2 - 8264*n + 2478)*x^3 - (624*n^3 - 3044*n^2 + 4434*n - 1706)*x^4 - (208*n^2 - 536*n + 248)*x^5)*D[P[n - 1, x], x] - ((20*n^4 - 86*n^3 + 136*n^2 - 91*n + 21)*x^2 - (80*n^4 - 404*n^3 + 766*n^2 - 615*n + 167)*x^3 - (240*n^3 - 948*n^2 + 1155*n - 401)*x^4 - (240*n^2 - 624*n + 288)*x^5 - (80*n - 48)*x^6)*D[P[n - 1, x], {x, 2}])/(16*n^6 - 128*n^5 + 420*n^4 - 712*n^3 + 650*n^2 - 300*n + 54 + (48*n^5 - 352*n^4 + 960*n^3 - 1198*n^2 + 684*n - 142)*x + (48*n^4 - 260*n^3 + 446*n^2 - 304*n + 70)*x^2 + (16*n^3 - 40*n^2 + 32*n - 8)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n^6 - 96*n^5 + 244*n^4 - 336*n^3 + 262*n^2 - 108*n + 18 + (144*n^6 - 1256*n^5 + 4400*n^4 - 7868*n^3 + 7558*n^2 - 3668*n + 690)*x + (432*n^5 - 3432*n^4 + 10396*n^3 - 15114*n^2 + 10440*n - 2666)*x^2 + (432*n^4 - 2684*n^3 + 5876*n^2 - 5304*n + 1604)*x^3 + (144*n^3 - 528*n^2 + 608*n - 208)*x^4)*v[i - 1] + ((32*n^5 - 216*n^4 + 548*n^3 - 660*n^2 + 374*n - 78)*x - (208*n^5 - 1584*n^4 + 4496*n^3 - 6052*n^2 + 3834*n - 890)*x^2 - (624*n^4 - 4048*n^3 + 9080*n^2 - 8264*n + 2478)*x^3 - (624*n^3 - 3044*n^2 + 4434*n - 1706)*x^4 - (208*n^2 - 536*n + 248)*x^5)*deriv(v[i - 1], x) - ((20*n^4 - 86*n^3 + 136*n^2 - 91*n + 21)*x^2 - (80*n^4 - 404*n^3 + 766*n^2 - 615*n + 167)*x^3 - (240*n^3 - 948*n^2 + 1155*n - 401)*x^4 - (240*n^2 - 624*n + 288)*x^5 - (80*n - 48)*x^6)*deriv(deriv(v[i - 1], x), x))/(16*n^6 - 128*n^5 + 420*n^4 - 712*n^3 + 650*n^2 - 300*n + 54 + (48*n^5 - 352*n^4 + 960*n^3 - 1198*n^2 + 684*n - 142)*x + (48*n^4 - 260*n^3 + 446*n^2 - 304*n + 70)*x^2 + (16*n^3 - 40*n^2 + 32*n - 8)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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