Real-rooted atlas

A200536 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(28 + 38\,x + 8\,x^{2}\right)\,P_{n-1}(x) + \left(38\,x + 16\,x^{2}\right)\,P_{n-1}'(x) + \left(8\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12838874
278435723236864648520
321952265544587480419256117856134405121426040
46146561836008085012592122906592732841122045568027827201761284096323596656
51721036812351222081119787260829450014560314764825601619047920043802722566474905605187584020889603276894648941888
6481890304821720597121410578995136638923482662411309793419584956620247520043349034450561124399368704172628917248157674700808321597442319974426214434407018489280
71349292851254416677725441737106257632001313352523195776368226181085593648207373430361603369402348178048136577828372262433798968831948852567621908480518938708377632167025049612025724928245891072209715215126778736728192

All 15 rows

Technical details
Node id
n-269219ae86b544b7 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-d1f860823afc60e5

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:             28,                38,                   8;
  2:            784,              3572,                3236,                  864,                    64;
  3:          21952,            265544,              587480,               419256,                117856,                 13440,                    512;
  4:         614656,          18360080,            85012592,            122906592,              73284112,              20455680,                2782720,                 176128,                  4096;
  5:       17210368,        1235122208,         11197872608,          29450014560,           31476482560,           16190479200,             4380272256,              647490560,              51875840,               2088960,                32768;
  6:      481890304,       82172059712,       1410578995136,        6389234826624,        11309793419584,         9566202475200,          4334903445056,          1124399368704,          172628917248,           15767470080,            832159744,            23199744,             262144;
  7:    13492928512,     5441667772544,     173710625763200,     1313352523195776,      3682261810855936,      4820737343036160,       3369402348178048,       1365778283722624,       337989688319488,        52567621908480,        5189387083776,        321670250496,        12025724928,         245891072,          2097152;
  8:   377801998336,   359662804271360,   21139233094729472,   261740256552801792,   1130642261797180672,   2208336240560540160,    2269528665661426176,    1350506132145400832,    495453918233018624,    116761484856238080,    18124853324398592,    1874574370013184,    128872146665472,     5774505934848,     160922861568,     2516582400,      16777216;
  9: 10578455953408, 23752101557846528, 2556132785214176768, 51213272364097574400, 334736040184320655360, 952170531687619023360, 1395315032648588567552, 1172602394374717953024, 607186205368856060928, 203505764157845521920, 45659056139621091328, 7008498451718307840, 744620847642836992, 54858064197844992, 2775926122217472, 94090611916800, 2029875363840, 25065160704, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (28+38*x+8*x^2)*P(n-1,x) + (38*x+16*x^2)*d/dx P(n-1,x) + 8*x^2*d^2/dx^2 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[(28 + 38*x + 8*x^2)*P[n - 1, x] + (38*x + 16*x^2)*D[P[n - 1, x], x] + 8*x^2*D[P[n - 1, x], {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] = (28 + 38*x + 8*x^2)*v[i - 1] + (38*x + 16*x^2)*deriv(v[i - 1], x) + 8*x^2*deriv(deriv(v[i - 1], x), 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

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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.