Real-rooted atlas

A054142 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(6 - 132\,x + 216\,x^{2} - 34\,n + 724\,n x - 1080\,n x^{2} + 68\,n^{2} - 1432\,n^{2} x + 2232\,n^{2} x^{2} - 56\,n^{3} + 1320\,n^{3} x - 2440\,n^{3} x^{2} + 16\,n^{4} - 576\,n^{4} x + 1488\,n^{4} x^{2} + 96\,n^{5} x - 480\,n^{5} x^{2} + 64\,n^{6} x^{2}\right)\,P_{n}(x) = \left(2 - 12\,x - 72\,x^{2} + 32\,x^{3} - 14\,n + 68\,n x + 456\,n x^{2} - 176\,n x^{3} + 36\,n^{2} - 168\,n^{2} x - 1124\,n^{2} x^{2} + 240\,n^{2} x^{3} - 40\,n^{3} + 280\,n^{3} x + 1744\,n^{3} x^{2} - 744\,n^{3} x^{3} + 16\,n^{4} - 288\,n^{4} x - 1024\,n^{4} x^{2} + 1296\,n^{4} x^{3} + 96\,n^{5} x + 64\,n^{5} x^{2} - 864\,n^{5} x^{3} + 64\,n^{6} x^{2} + 192\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(x - 6\,x^{2} - 36\,x^{3} + 16\,x^{4} - 6\,n x + 28\,n x^{2} + 188\,n x^{3} - 80\,n x^{4} + 12\,n^{2} x - 56\,n^{2} x^{2} - 396\,n^{2} x^{3} + 80\,n^{2} x^{4} - 8\,n^{3} x + 88\,n^{3} x^{2} + 424\,n^{3} x^{3} - 224\,n^{3} x^{4} - 48\,n^{4} x^{2} - 112\,n^{4} x^{3} + 320\,n^{4} x^{4} - 32\,n^{5} x^{3} - 128\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
23811
35482780
4712027064461
592249459601252263
6113602268448026252169960
71352844551344015750604834340577
815728772231680577504536012348512156115
917960122856406416087519958411319023040729574744
10191224183601164803753756486485885882534404009510002043229
11211520261631958407735001729728220720515375365212356600013317060079
1223184835910310080145350040098246670664658944036486451001000103818172823826480
13252208478174681602543625837216017283084224040961772199080080001816815157248219778827425
1427260062100680064420612516116408398840406449664067343562437580001647245631449602306852744256400411
Technical details
Node id
n-4b23c4d768293017 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c6ddcaa2b6ee4048

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 -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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  1;
  2:  3,   8;
  3:  5,  48,    27;
  4:  7, 120,   270,    64;
  5:  9, 224,   945,   960,    125;
  6: 11, 360,  2268,  4480,   2625,    216;
  7: 13, 528,  4455, 13440,  15750,   6048,    343;
  8: 15, 728,  7722, 31680,  57750,  45360,  12348,   512;
  9: 17, 960, 12285, 64064, 160875, 199584, 113190, 23040, 729;
%F Formula
The row polynomials satisfy (16*n^4-56*n^3+68*n^2-34*n+6+(96*n^5-576*n^4+1320*n^3-1432*n^2+724*n-132)*x+(64*n^6-480*n^5+1488*n^4-2440*n^3+2232*n^2-1080*n+216)*x^2)*P(n,x) = (16*n^4-40*n^3+36*n^2-14*n+2+(96*n^5-288*n^4+280*n^3-168*n^2+68*n-12)*x+(64*n^6+64*n^5-1024*n^4+1744*n^3-1124*n^2+456*n-72)*x^2+(192*n^6-864*n^5+1296*n^4-744*n^3+240*n^2-176*n+32)*x^3)*P(n-1,x) - ((8*n^3-12*n^2+6*n-1)*x+(48*n^4-88*n^3+56*n^2-28*n+6)*x^2+(32*n^5+112*n^4-424*n^3+396*n^2-188*n+36)*x^3+(128*n^5-320*n^4+224*n^3-80*n^2+80*n-16)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
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 - 288*n^4 + 280*n^3 - 168*n^2 + 68*n - 12)*x + (64*n^6 + 64*n^5 - 1024*n^4 + 1744*n^3 - 1124*n^2 + 456*n - 72)*x^2 + (192*n^6 - 864*n^5 + 1296*n^4 - 744*n^3 + 240*n^2 - 176*n + 32)*x^3)*P[n - 1, x] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (48*n^4 - 88*n^3 + 56*n^2 - 28*n + 6)*x^2 + (32*n^5 + 112*n^4 - 424*n^3 + 396*n^2 - 188*n + 36)*x^3 + (128*n^5 - 320*n^4 + 224*n^3 - 80*n^2 + 80*n - 16)*x^4)*D[P[n - 1, x], x])/(16*n^4 - 56*n^3 + 68*n^2 - 34*n + 6 + (96*n^5 - 576*n^4 + 1320*n^3 - 1432*n^2 + 724*n - 132)*x + (64*n^6 - 480*n^5 + 1488*n^4 - 2440*n^3 + 2232*n^2 - 1080*n + 216)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; 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 - 288*n^4 + 280*n^3 - 168*n^2 + 68*n - 12)*x + (64*n^6 + 64*n^5 - 1024*n^4 + 1744*n^3 - 1124*n^2 + 456*n - 72)*x^2 + (192*n^6 - 864*n^5 + 1296*n^4 - 744*n^3 + 240*n^2 - 176*n + 32)*x^3)*v[i - 1] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (48*n^4 - 88*n^3 + 56*n^2 - 28*n + 6)*x^2 + (32*n^5 + 112*n^4 - 424*n^3 + 396*n^2 - 188*n + 36)*x^3 + (128*n^5 - 320*n^4 + 224*n^3 - 80*n^2 + 80*n - 16)*x^4)*deriv(v[i - 1], x))/(16*n^4 - 56*n^3 + 68*n^2 - 34*n + 6 + (96*n^5 - 576*n^4 + 1320*n^3 - 1432*n^2 + 724*n - 132)*x + (64*n^6 - 480*n^5 + 1488*n^4 - 2440*n^3 + 2232*n^2 - 1080*n + 216)*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