Real-rooted atlas

A086645 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(10 - 590\,x + 1000\,x^{2} - 9\,n + 1021\,n x - 2700\,n x^{2} + 2\,n^{2} - 659\,n^{2} x + 3030\,n^{2} x^{2} + 188\,n^{3} x - 1809\,n^{3} x^{2} - 20\,n^{4} x + 606\,n^{4} x^{2} - 108\,n^{5} x^{2} + 8\,n^{6} x^{2}\right)\,P_{n}(x) = \left(3 - 109\,x - 541\,x^{2} + 227\,x^{3} - 5\,n + 350\,n x + 1200\,n x^{2} - 1117\,n x^{3} + 2\,n^{2} - 369\,n^{2} x - 632\,n^{2} x^{2} + 2115\,n^{2} x^{3} + 148\,n^{3} x - 269\,n^{3} x^{2} - 1987\,n^{3} x^{3} - 20\,n^{4} x + 326\,n^{4} x^{2} + 982\,n^{4} x^{3} - 92\,n^{5} x^{2} - 244\,n^{5} x^{3} + 8\,n^{6} x^{2} + 24\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x + 102\,x^{2} - 132\,x^{3} + 34\,x^{4} + 4\,n x - 238\,n x^{2} + 420\,n x^{3} - 186\,n x^{4} + 176\,n^{2} x^{2} - 524\,n^{2} x^{3} + 348\,n^{2} x^{4} - 40\,n^{3} x^{2} + 332\,n^{3} x^{3} - 292\,n^{3} x^{4} - 112\,n^{4} x^{3} + 112\,n^{4} x^{4} + 16\,n^{5} x^{3} - 16\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
36814
41512027162
528560756641408
64516805670288012510400
76639602494831680825021669120
89180088108119219212512519656343426496
9120145602162168236801001000393120411605122490368
101532448050122828005125469750400982410495807833672913934592
1119038760104652080620802309450027209520132946802480640138510100075366400
1223158520201455120465280808307501396755361096811103820185653326352310001331396492800
1327685008363409247070144245157000584097696672710808376561152981204841062600036735617282038431744
14325119600621621099985600663966875208606320033125911002719608320113889847523023000019898450561600219710278141952
15378163800101719801989187201640388750657109908013759993800155759385609566747190310810500050144094035380800830466274450969182208
Technical details
Node id
n-fbd62aba6d47a176 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2a9ba3be646fcc5b

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:   0;
   2:   1;
   3:   6,     8;
   4:  15,   120,     27;
   5:  28,   560,    756,      64;
   6:  45,  1680,   5670,    2880,     125;
   7:  66,  3960,  24948,   31680,    8250,     216;
   8:  91,  8008,  81081,  192192,  125125,   19656,     343;
   9: 120, 14560, 216216,  823680, 1001000,  393120,   41160,   512;
  10: 153, 24480, 501228, 2800512, 5469750, 4009824, 1049580, 78336, 729;
%F Formula
The row polynomials satisfy (2*n^2-9*n+10-(20*n^4-188*n^3+659*n^2-1021*n+590)*x+(8*n^6-108*n^5+606*n^4-1809*n^3+3030*n^2-2700*n+1000)*x^2)*P(n,x) = (2*n^2-5*n+3-(20*n^4-148*n^3+369*n^2-350*n+109)*x+(8*n^6-92*n^5+326*n^4-269*n^3-632*n^2+1200*n-541)*x^2+(24*n^6-244*n^5+982*n^4-1987*n^3+2115*n^2-1117*n+227)*x^3)*P(n-1,x) + ((4*n-4)*x-(40*n^3-176*n^2+238*n-102)*x^2+(16*n^5-112*n^4+332*n^3-524*n^2+420*n-132)*x^3-(16*n^5-112*n^4+292*n^3-348*n^2+186*n-34)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3 - (20*n^4 - 148*n^3 + 369*n^2 - 350*n + 109)*x + (8*n^6 - 92*n^5 + 326*n^4 - 269*n^3 - 632*n^2 + 1200*n - 541)*x^2 + (24*n^6 - 244*n^5 + 982*n^4 - 1987*n^3 + 2115*n^2 - 1117*n + 227)*x^3)*P[n - 1, x] + ((4*n - 4)*x - (40*n^3 - 176*n^2 + 238*n - 102)*x^2 + (16*n^5 - 112*n^4 + 332*n^3 - 524*n^2 + 420*n - 132)*x^3 - (16*n^5 - 112*n^4 + 292*n^3 - 348*n^2 + 186*n - 34)*x^4)*D[P[n - 1, x], x])/(2*n^2 - 9*n + 10 - (20*n^4 - 188*n^3 + 659*n^2 - 1021*n + 590)*x + (8*n^6 - 108*n^5 + 606*n^4 - 1809*n^3 + 3030*n^2 - 2700*n + 1000)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((2*n^2 - 5*n + 3 - (20*n^4 - 148*n^3 + 369*n^2 - 350*n + 109)*x + (8*n^6 - 92*n^5 + 326*n^4 - 269*n^3 - 632*n^2 + 1200*n - 541)*x^2 + (24*n^6 - 244*n^5 + 982*n^4 - 1987*n^3 + 2115*n^2 - 1117*n + 227)*x^3)*v[i - 1] + ((4*n - 4)*x - (40*n^3 - 176*n^2 + 238*n - 102)*x^2 + (16*n^5 - 112*n^4 + 332*n^3 - 524*n^2 + 420*n - 132)*x^3 - (16*n^5 - 112*n^4 + 292*n^3 - 348*n^2 + 186*n - 34)*x^4)*deriv(v[i - 1], x))/(2*n^2 - 9*n + 10 - (20*n^4 - 188*n^3 + 659*n^2 - 1021*n + 590)*x + (8*n^6 - 108*n^5 + 606*n^4 - 1809*n^3 + 3030*n^2 - 2700*n + 1000)*x^2)); 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

Same fingerprint ?

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