Real-rooted atlas

A125662 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(128\,n + 32\,n x + 32\,n x^{2} - 240\,n^{3} x^{2} - 120\,n^{3} x^{3} + 18\,n^{5} x^{4}\right)\,P_{n}(x) = \left(256 + 496\,x + 304\,x^{2} - 24\,x^{3} - 72\,x^{4} + 256\,n + 832\,n x + 256\,n x^{2} - 636\,n x^{3} - 396\,n x^{4} + 336\,n^{2} x - 528\,n^{2} x^{2} - 1392\,n^{2} x^{3} - 648\,n^{2} x^{4} + 27\,n^{2} x^{5} - 480\,n^{3} x^{2} - 852\,n^{3} x^{3} - 324\,n^{3} x^{4} + 81\,n^{3} x^{5} - 72\,n^{4} x^{3} + 36\,n^{4} x^{4} + 81\,n^{4} x^{5} + 36\,n^{5} x^{4} + 27\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(64 + 192\,x + 144\,x^{2} - 64\,x^{3} - 120\,x^{4} - 36\,x^{5} - 336\,n x^{2} - 912\,n x^{3} - 732\,n x^{4} - 180\,n x^{5} - 336\,n^{2} x^{2} - 912\,n^{2} x^{3} - 696\,n^{2} x^{4} - 108\,n^{2} x^{5} + 27\,n^{2} x^{6} + 72\,n^{3} x^{4} + 144\,n^{3} x^{5} + 54\,n^{3} x^{6} + 36\,n^{4} x^{4} + 72\,n^{4} x^{5} + 27\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141216
2159681192
3565529724322012
4209278475336912202519463
5780130924795267392405008748178464
6291158944272646518400475875209952357211574449
7108642576161440504346032042930002886840100018813996813489300

All 15 rows

Technical details
Node id
n-a58ac64aaef49d72 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-287cc83da219a575

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,      12;
  2:     15,      96,       81;
  3:     56,     552,      972,       432;
  4:    209,    2784,     7533,      6912,      2025;
  5:    780,   13092,    47952,     67392,     40500,      8748;
  6:   2911,   58944,   272646,    518400,    475875,    209952,     35721;
  7:  10864,  257616,  1440504,   3460320,   4293000,   2886840,   1000188,   139968;
  8:  40545, 1101888,  7228278,  21005568,  32916375,  29883168,  15752961,  4478976,   531441;
  9: 151316, 4636476, 34910352, 119112768, 225836100, 258844572, 184034592, 79501824, 19131876, 1968300;
%F Formula
The row polynomials satisfy (128*n+32*n*x-(240*n^3-32*n)*x^2-120*n^3*x^3+18*n^5*x^4)*P(n,x) = (256*n+256+(336*n^2+832*n+496)*x-(480*n^3+528*n^2-256*n-304)*x^2-(72*n^4+852*n^3+1392*n^2+636*n+24)*x^3+(36*n^5+36*n^4-324*n^3-648*n^2-396*n-72)*x^4+(27*n^5+81*n^4+81*n^3+27*n^2)*x^5)*P(n-1,x) + (64+192*x-(336*n^2+336*n-144)*x^2-(912*n^2+912*n+64)*x^3+(36*n^4+72*n^3-696*n^2-732*n-120)*x^4+(72*n^4+144*n^3-108*n^2-180*n-36)*x^5+(27*n^4+54*n^3+27*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[((256*n + 256 + (336*n^2 + 832*n + 496)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (72*n^4 + 852*n^3 + 1392*n^2 + 636*n + 24)*x^3 + (36*n^5 + 36*n^4 - 324*n^3 - 648*n^2 - 396*n - 72)*x^4 + (27*n^5 + 81*n^4 + 81*n^3 + 27*n^2)*x^5)*P[n - 1, x] + (64 + 192*x - (336*n^2 + 336*n - 144)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (36*n^4 + 72*n^3 - 696*n^2 - 732*n - 120)*x^4 + (72*n^4 + 144*n^3 - 108*n^2 - 180*n - 36)*x^5 + (27*n^4 + 54*n^3 + 27*n^2)*x^6)*D[P[n - 1, x], x])/(128*n + 32*n*x - (240*n^3 - 32*n)*x^2 - 120*n^3*x^3 + 18*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] = ((256*n + 256 + (336*n^2 + 832*n + 496)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (72*n^4 + 852*n^3 + 1392*n^2 + 636*n + 24)*x^3 + (36*n^5 + 36*n^4 - 324*n^3 - 648*n^2 - 396*n - 72)*x^4 + (27*n^5 + 81*n^4 + 81*n^3 + 27*n^2)*x^5)*v[i - 1] + (64 + 192*x - (336*n^2 + 336*n - 144)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (36*n^4 + 72*n^3 - 696*n^2 - 732*n - 120)*x^4 + (72*n^4 + 144*n^3 - 108*n^2 - 180*n - 36)*x^5 + (27*n^4 + 54*n^3 + 27*n^2)*x^6)*deriv(v[i - 1], x))/(128*n + 32*n*x - (240*n^3 - 32*n)*x^2 - 120*n^3*x^3 + 18*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