Real-rooted atlas

A038334 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(36 + 16\,x\right)\,P_{n-1}(x) + \left(16\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1296 + 1408\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0129614082562960
14665693952399364096184640
21679616563200042188809830406553612579072
360466176319737856376995840150077440222822401048576930608128
4217678233617593827328307515228161863843840046294630404781506561677721674284961792
5783641640969497075384322372604788736205765319065676116131840012953688473698650030082684354566359161323520
62821109907456506386186240001765324462489602108045446348801090385828904962720485972377633747705528321975684956164294967296580616796045312
7101559956668416267834592696729612814424242913280205321008893132801427673100320768049003810542059528807474685542408323646619648038654705664006871947673656271461198069760

All 15 rows

Technical details
Node id
n-1bf987212f59a9c9 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-65e248b1aaac025a

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:               1296,                1408,                  256;
  1:              46656,               93952,                39936,                  4096;
  2:            1679616,             5632000,              4218880,                983040,                 65536;
  3:           60466176,           319737856,            376995840,             150077440,              22282240,               1048576;
  4:         2176782336,         17593827328,          30751522816,           18638438400,            4629463040,             478150656,             16777216;
  5:        78364164096,        949707538432,        2372604788736,         2057653190656,          761161318400,          129536884736,           9865003008,           268435456;
  6:      2821109907456,      50638618624000,      176532446248960,       210804544634880,       109038582890496,        27204859723776,        3374770552832,        197568495616,         4294967296;
  7:    101559956668416,    2678345926967296,    12814424242913280,     20532100889313280,     14276731003207680,      4900381054205952,      880747468554240,      83236466196480,      3865470566400,       68719476736;
  8:   3656158440062976,  140898947508994048,   914234383349579776,   1929727262588928000,   1756186714549780480,    796871898339213312,   194664762716454912,   26410956493946880,   1965720632033280,    74217034874880,    1099511627776;
  9: 131621703842267136, 7385243805508698112, 64422321227915329536, 176724840191063228416, 206494307656400896000, 120536127640145231872, 38445699051999461376, 7023457764567416832, 744953487556608000, 44810596390010880, 1402976837042176, 17592186044416;
%F Formula
The row polynomials satisfy P(n,x) = (36+16*x)*P(n-1,x) + 16*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1296 + 1408*x + 256*x^2.
%t Mathematica
P[0, x_] := 1296 + 1408*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(36 + 16*x)*P[n - 1, x] + 16*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1296 + 1408*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 16*x)*v[i - 1] + 16*x*deriv(v[i - 1], x)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.