Real-rooted atlas

A086645 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-18 + 156\,x - 72\,x^{2} + 33\,n - 398\,n x + 276\,n x^{2} - 20\,n^{2} + 376\,n^{2} x - 416\,n^{2} x^{2} + 4\,n^{3} - 156\,n^{3} x + 308\,n^{3} x^{2} + 24\,n^{4} x - 112\,n^{4} x^{2} + 16\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-72 - 240\,x + 160\,x^{2} + 64\,x^{3} + 132\,n - 8\,n x - 1984\,n x^{2} - 480\,n x^{3} - 80\,n^{2} + 544\,n^{2} x + 2904\,n^{2} x^{2} + 16\,n^{3} - 432\,n^{3} x - 1296\,n^{3} x^{2} + 1008\,n^{3} x^{3} + 96\,n^{4} x + 32\,n^{4} x^{2} - 832\,n^{4} x^{3} + 64\,n^{5} x^{2} + 192\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-18\,x - 68\,x^{2} + 24\,x^{3} + 32\,x^{4} + 24\,n x - 16\,n x^{2} - 504\,n x^{3} - 224\,n x^{4} - 8\,n^{2} x + 112\,n^{2} x^{2} + 584\,n^{2} x^{3} + 32\,n^{2} x^{4} - 48\,n^{3} x^{2} - 128\,n^{3} x^{3} + 256\,n^{3} x^{4} - 32\,n^{4} x^{3} - 128\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
221618
38128144280
432768259210244416
512840962304032768640066432
65122048016128040960032000036864948736
720489830499532836700165376000265420820070412996608
881924587525677056275251206021120057802752196689921048576172400640

All 15 rows

Technical details
Node id
n-148975a3f35d4c6f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-31dd68f9934466a8

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:      2,      16;
   3:      8,     128,       144;
   4:     32,     768,      2592,       1024;
   5:    128,    4096,     23040,      32768,       6400;
   6:    512,   20480,    161280,     409600,     320000,      36864;
   7:   2048,   98304,    995328,    3670016,    5376000,    2654208,     200704;
   8:   8192,  458752,   5677056,   27525120,   60211200,   57802752,   19668992,    1048576;
   9:  32768, 2097152,  30670848,  184549376,  540672000,  792723456,  539492352,  134217728,   5308416;
  10: 131072, 9437184, 159252480, 1145044992, 4217241600, 8408530944, 8901623808, 4529848320, 859963392, 26214400;
%F Formula
The row polynomials satisfy (4*n^3-20*n^2+33*n-18+(24*n^4-156*n^3+376*n^2-398*n+156)*x+(16*n^5-112*n^4+308*n^3-416*n^2+276*n-72)*x^2)*P(n,x) = (16*n^3-80*n^2+132*n-72+(96*n^4-432*n^3+544*n^2-8*n-240)*x+(64*n^5+32*n^4-1296*n^3+2904*n^2-1984*n+160)*x^2+(192*n^5-832*n^4+1008*n^3-480*n+64)*x^3)*P(n-1,x) - ((8*n^2-24*n+18)*x+(48*n^3-112*n^2+16*n+68)*x^2+(32*n^4+128*n^3-584*n^2+504*n-24)*x^3+(128*n^4-256*n^3-32*n^2+224*n-32)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 16*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 80*n^2 + 132*n - 72 + (96*n^4 - 432*n^3 + 544*n^2 - 8*n - 240)*x + (64*n^5 + 32*n^4 - 1296*n^3 + 2904*n^2 - 1984*n + 160)*x^2 + (192*n^5 - 832*n^4 + 1008*n^3 - 480*n + 64)*x^3)*P[n - 1, x] - ((8*n^2 - 24*n + 18)*x + (48*n^3 - 112*n^2 + 16*n + 68)*x^2 + (32*n^4 + 128*n^3 - 584*n^2 + 504*n - 24)*x^3 + (128*n^4 - 256*n^3 - 32*n^2 + 224*n - 32)*x^4)*D[P[n - 1, x], x])/(4*n^3 - 20*n^2 + 33*n - 18 + (24*n^4 - 156*n^3 + 376*n^2 - 398*n + 156)*x + (16*n^5 - 112*n^4 + 308*n^3 - 416*n^2 + 276*n - 72)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 16*x; for(n = 3, N, my(i = n); v[i] = ((16*n^3 - 80*n^2 + 132*n - 72 + (96*n^4 - 432*n^3 + 544*n^2 - 8*n - 240)*x + (64*n^5 + 32*n^4 - 1296*n^3 + 2904*n^2 - 1984*n + 160)*x^2 + (192*n^5 - 832*n^4 + 1008*n^3 - 480*n + 64)*x^3)*v[i - 1] - ((8*n^2 - 24*n + 18)*x + (48*n^3 - 112*n^2 + 16*n + 68)*x^2 + (32*n^4 + 128*n^3 - 584*n^2 + 504*n - 24)*x^3 + (128*n^4 - 256*n^3 - 32*n^2 + 224*n - 32)*x^4)*deriv(v[i - 1], x))/(4*n^3 - 20*n^2 + 33*n - 18 + (24*n^4 - 156*n^3 + 376*n^2 - 398*n + 156)*x + (16*n^5 - 112*n^4 + 308*n^3 - 416*n^2 + 276*n - 72)*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