Real-rooted atlas

A125662 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(72\,n + 24\,n x + 32\,n x^{2} - 240\,n^{3} x^{2} - 160\,n^{3} x^{3} + 32\,n^{5} x^{4}\right)\,P_{n}(x) = \left(144 + 372\,x + 304\,x^{2} - 32\,x^{3} - 128\,x^{4} + 144\,n + 624\,n x + 256\,n x^{2} - 848\,n x^{3} - 704\,n x^{4} + 252\,n^{2} x - 528\,n^{2} x^{2} - 1856\,n^{2} x^{3} - 1152\,n^{2} x^{4} + 64\,n^{2} x^{5} - 480\,n^{3} x^{2} - 1136\,n^{3} x^{3} - 576\,n^{3} x^{4} + 192\,n^{3} x^{5} - 96\,n^{4} x^{3} + 64\,n^{4} x^{4} + 192\,n^{4} x^{5} + 64\,n^{5} x^{4} + 64\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(27 + 108\,x + 108\,x^{2} - 64\,x^{3} - 160\,x^{4} - 64\,x^{5} - 252\,n x^{2} - 912\,n x^{3} - 976\,n x^{4} - 320\,n x^{5} - 252\,n^{2} x^{2} - 912\,n^{2} x^{3} - 928\,n^{2} x^{4} - 192\,n^{2} x^{5} + 64\,n^{2} x^{6} + 96\,n^{3} x^{4} + 256\,n^{3} x^{5} + 128\,n^{3} x^{6} + 48\,n^{4} x^{4} + 128\,n^{4} x^{5} + 64\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141620
215128144287
356736172810243544
420937121339216384640040097
5780174568524815974412800036864428092
6291178592484704122880015040008847362007044384447
7108643434882560896820224013568000121651205619712104857643518896

All 15 rows

Technical details
Node id
n-8218aca39de1d09b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-59b05671af16d5d5

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,      16;
  2:     15,     128,      144;
  3:     56,     736,     1728,      1024;
  4:    209,    3712,    13392,     16384,      6400;
  5:    780,   17456,    85248,    159744,    128000,      36864;
  6:   2911,   78592,   484704,   1228800,   1504000,     884736,     200704;
  7:  10864,  343488,  2560896,   8202240,  13568000,   12165120,    5619712,   1048576;
  8:  40545, 1469184, 12850272,  49790976, 104032000,  125927424,   88510464,  33554432,   5308416;
  9: 151316, 6181968, 62062848, 282341376, 713753600, 1090768896, 1034027008, 595591168, 191102976, 26214400;
%F Formula
The row polynomials satisfy (72*n+24*n*x-(240*n^3-32*n)*x^2-160*n^3*x^3+32*n^5*x^4)*P(n,x) = (144*n+144+(252*n^2+624*n+372)*x-(480*n^3+528*n^2-256*n-304)*x^2-(96*n^4+1136*n^3+1856*n^2+848*n+32)*x^3+(64*n^5+64*n^4-576*n^3-1152*n^2-704*n-128)*x^4+(64*n^5+192*n^4+192*n^3+64*n^2)*x^5)*P(n-1,x) + (27+108*x-(252*n^2+252*n-108)*x^2-(912*n^2+912*n+64)*x^3+(48*n^4+96*n^3-928*n^2-976*n-160)*x^4+(128*n^4+256*n^3-192*n^2-320*n-64)*x^5+(64*n^4+128*n^3+64*n^2)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((144*n + 144 + (252*n^2 + 624*n + 372)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (96*n^4 + 1136*n^3 + 1856*n^2 + 848*n + 32)*x^3 + (64*n^5 + 64*n^4 - 576*n^3 - 1152*n^2 - 704*n - 128)*x^4 + (64*n^5 + 192*n^4 + 192*n^3 + 64*n^2)*x^5)*P[n - 1, x] + (27 + 108*x - (252*n^2 + 252*n - 108)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (48*n^4 + 96*n^3 - 928*n^2 - 976*n - 160)*x^4 + (128*n^4 + 256*n^3 - 192*n^2 - 320*n - 64)*x^5 + (64*n^4 + 128*n^3 + 64*n^2)*x^6)*D[P[n - 1, x], x])/(72*n + 24*n*x - (240*n^3 - 32*n)*x^2 - 160*n^3*x^3 + 32*n^5*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((144*n + 144 + (252*n^2 + 624*n + 372)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (96*n^4 + 1136*n^3 + 1856*n^2 + 848*n + 32)*x^3 + (64*n^5 + 64*n^4 - 576*n^3 - 1152*n^2 - 704*n - 128)*x^4 + (64*n^5 + 192*n^4 + 192*n^3 + 64*n^2)*x^5)*v[i - 1] + (27 + 108*x - (252*n^2 + 252*n - 108)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (48*n^4 + 96*n^3 - 928*n^2 - 976*n - 160)*x^4 + (128*n^4 + 256*n^3 - 192*n^2 - 320*n - 64)*x^5 + (64*n^4 + 128*n^3 + 64*n^2)*x^6)*deriv(v[i - 1], x))/(72*n + 24*n*x - (240*n^3 - 32*n)*x^2 - 160*n^3*x^3 + 32*n^5*x^4)); 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