Real-rooted atlas

A123519 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - 15\,x + 18\,x^{2} - 4\,n - 10\,n x + 39\,n x^{2} + 27\,n^{2} x^{2} + 6\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-14 - 95\,x - 78\,x^{2} + 108\,x^{3} - 8\,n - 62\,n x + 27\,n x^{2} + 198\,n x^{3} - 6\,n^{2} x + 57\,n^{2} x^{2} + 108\,n^{2} x^{3} + 12\,n^{3} x^{2} + 18\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-2 - 18\,x - 28\,x^{2} + 42\,x^{3} + 54\,x^{4} + 24\,n x^{2} + 96\,n x^{3} + 72\,n x^{4} + 6\,n^{2} x^{2} + 24\,n^{2} x^{3} + 18\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 11 + 84\,x + 108\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01184108203
14145611888642549
21532232874812960648030573
3571103085389212096012312046656355507
4213145852299592902016139320010730883265924042471
579531986721557144588384012214800142767368817984223948845196617
629681844272771703235090496915948001439804161348824966942412815116544498679865
71107713534588369094321962144006173431201221127488153857491212003655685290790401007769605444036279

All 15 rows

Technical details
Node id
n-b2b0d5af4571163f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7a5ba09a54c4c750

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      11,       84,       108;
  1:      41,      456,      1188,        864;
  2:     153,     2232,      8748,      12960,        6480;
  3:     571,    10308,     53892,     120960,      123120,       46656;
  4:    2131,    45852,    299592,     902016,     1393200,     1073088,       326592;
  5:    7953,   198672,   1557144,    5883840,    12214800,    14276736,      8817984,      2239488;
  6:   29681,   844272,   7717032,   35090496,    91594800,   143980416,    134882496,     69424128,     15116544;
  7:  110771,  3534588,  36909432,  196214400,   617343120,  1221127488,   1538574912,   1200365568,    529079040,   100776960;
  8:  413403, 14623332, 171731988, 1045042560,  3849385680,  9185400000,  14567309568,  15282266112,  10203667200,  3930301440,   665127936;
  9: 1542841, 59919576, 781628508, 5357811744, 22618018800, 63236049408, 121028463360, 159818821632, 143440886016, 83644876800, 28600501248, 4353564672;
%F Formula
The row polynomials satisfy -(4*n+6+(10*n+15)*x-(6*n^3+27*n^2+39*n+18)*x^2)*P(n,x) = -(8*n+14+(6*n^2+62*n+95)*x-(12*n^3+57*n^2+27*n-78)*x^2-(18*n^3+108*n^2+198*n+108)*x^3)*P(n-1,x) - (2+18*x-(6*n^2+24*n-28)*x^2-(24*n^2+96*n+42)*x^3-(18*n^2+72*n+54)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 84*x + 108*x^2.
%t Mathematica
P[0, x_] := 11 + 84*x + 108*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n + 14 + (6*n^2 + 62*n + 95)*x - (12*n^3 + 57*n^2 + 27*n - 78)*x^2 - (18*n^3 + 108*n^2 + 198*n + 108)*x^3)*P[n - 1, x] - (2 + 18*x - (6*n^2 + 24*n - 28)*x^2 - (24*n^2 + 96*n + 42)*x^3 - (18*n^2 + 72*n + 54)*x^4)*D[P[n - 1, x], x])/(-4*n - 6 - (10*n + 15)*x + (6*n^3 + 27*n^2 + 39*n + 18)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 11 + 84*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n + 14 + (6*n^2 + 62*n + 95)*x - (12*n^3 + 57*n^2 + 27*n - 78)*x^2 - (18*n^3 + 108*n^2 + 198*n + 108)*x^3)*v[i - 1] - (2 + 18*x - (6*n^2 + 24*n - 28)*x^2 - (24*n^2 + 96*n + 42)*x^3 - (18*n^2 + 72*n + 54)*x^4)*deriv(v[i - 1], x))/(-4*n - 6 - (10*n + 15)*x + (6*n^3 + 27*n^2 + 39*n + 18)*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