Real-rooted atlas

A114687 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-28 - 208\,x + 128\,x^{2} - 78\,n + 152\,n x + 64\,n x^{2} + 96\,n^{2} + 128\,n^{2} x + 64\,n^{3}\right)\,P_{n}(x) = \left(-28 - 208\,x + 128\,x^{2} - 114\,n - 1256\,n x - 512\,n x^{2} + 1536\,n x^{3} + 288\,n^{2} + 1152\,n^{2} x + 3072\,n^{2} x^{2} + 448\,n^{3} + 1536\,n^{3} x\right)\,P_{n-1}(x) + \left(-5 - 72\,x - 336\,x^{2} - 768\,x^{3} - 1024\,x^{4} - 32\,n - 448\,n x - 1792\,n x^{2} - 2048\,n x^{3} - 32\,n^{2} - 384\,n^{2} x - 1024\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111819
26718464315
3381187223045125069
42133152483430425600409681381
511903112120373120517120266240327681313271
666463778344345312071756806942720265420826214421332679
73718015210912289873928110592011898880086245376256901122097152348697465

All 15 rows

Technical details
Node id
n-70f1c1b6f10e9bd4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bed5ea9fedc7f625

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:       11,         8;
  2:       67,       184,         64;
  3:      381,      1872,       2304,        512;
  4:     2133,     15248,      34304,      25600,        4096;
  5:    11903,    112120,     373120,     517120,      266240,       32768;
  6:    66463,    778344,    3453120,    7175680,     6942720,     2654208,      262144;
  7:   371801,   5210912,   28987392,   81105920,   118988800,    86245376,    25690112,     2097152;
  8:  2084521,  34036768,  228057088,  806055936,  1605976064,  1781792768,  1012924416,   243269632,   16777216;
  9: 11713203, 218420712, 1713574656, 7336734720, 18586681344, 28216000512, 24750587904, 11400118272, 2264924160, 134217728;
%F Formula
The row polynomials satisfy (64*n^3+96*n^2-78*n-28+(128*n^2+152*n-208)*x+(64*n+128)*x^2)*P(n,x) = (448*n^3+288*n^2-114*n-28+(1536*n^3+1152*n^2-1256*n-208)*x+(3072*n^2-512*n+128)*x^2+1536*n*x^3)*P(n-1,x) - (32*n^2+32*n+5+(384*n^2+448*n+72)*x+(1024*n^2+1792*n+336)*x^2+(2048*n+768)*x^3+1024*x^4)*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[((448*n^3 + 288*n^2 - 114*n - 28 + (1536*n^3 + 1152*n^2 - 1256*n - 208)*x + (3072*n^2 - 512*n + 128)*x^2 + 1536*n*x^3)*P[n - 1, x] - (32*n^2 + 32*n + 5 + (384*n^2 + 448*n + 72)*x + (1024*n^2 + 1792*n + 336)*x^2 + (2048*n + 768)*x^3 + 1024*x^4)*D[P[n - 1, x], x])/(64*n^3 + 96*n^2 - 78*n - 28 + (128*n^2 + 152*n - 208)*x + (64*n + 128)*x^2)]];
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] = ((448*n^3 + 288*n^2 - 114*n - 28 + (1536*n^3 + 1152*n^2 - 1256*n - 208)*x + (3072*n^2 - 512*n + 128)*x^2 + 1536*n*x^3)*v[i - 1] - (32*n^2 + 32*n + 5 + (384*n^2 + 448*n + 72)*x + (1024*n^2 + 1792*n + 336)*x^2 + (2048*n + 768)*x^3 + 1024*x^4)*deriv(v[i - 1], x))/(64*n^3 + 96*n^2 - 78*n - 28 + (128*n^2 + 152*n - 208)*x + (64*n + 128)*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