Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 16\,x + 4\,x^{2}\right)\,P_{n-1}(x) + \left(16\,x + 8\,x^{2}\right)\,P_{n-1}'(x) + \left(4\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1916429
281544616192161449
372914896435643744012080153664110309
465613840642495056448627231335369891841487361024025611653905
55904997065761315015244447660805685889603355269121013940481643520014246406144010241609465453
6531441243609184667676869640272927552871159134568577538252843877523712125810995202103205632206807040116469763440644096278865763897
7478296960987326563330104122843476738166720121786497050401883869520332815024327910592678720080896018406064934403097827532803277309235221603778568535654418350081638458830135647413

All 15 rows

Technical details
Node id
n-1f22c0bcbe735965 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-14e66869ea533ba5

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:         9,            16,               4;
  2:        81,           544,             616,               192,                 16;
  3:       729,         14896,           43564,             37440,              12080,               1536,                  64;
  4:      6561,        384064,         2495056,           4486272,            3133536,             989184,              148736,               10240,                256;
  5:     59049,       9706576,       131501524,         444766080,          568588960,          335526912,           101394048,            16435200,            1424640,              61440,              1024;
  6:    531441,     243609184,      6676768696,       40272927552,        87115913456,        85775382528,         43877523712,         12581099520,         2103205632,          206807040,          11646976,           344064,             4096;
  7:   4782969,    6098732656,    333010412284,     3476738166720,     12178649705040,     18838695203328,      15024327910592,       6787200808960,      1840606493440,       309782753280,       32773092352,       2160377856,         85356544,        1835008,         16384;
  8:  43046721,  152544843904,  16463898917536,   292296519628032,   1614018182627776,   3782221627871232,    4484155310089728,    2998412800880640,   1213234714019328,    311011516907520,    51960274116608,    5735584038912,     417996587008,    19744686080,     577372160,      9437184,      65536;
  9: 387420489, 3814309845136, 810392295399844, 24203473034607360, 207053916478752320, 717837513954883584, 1227195428661488896, 1168656125811732480, 671800074664742400, 246208507281285120, 59708937351624704, 9816972407144448, 1109061348442112, 86363234893824, 4596203323392, 163159474176, 3673227264, 47185920, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (9+16*x+4*x^2)*P(n-1,x) + (16*x+8*x^2)*d/dx P(n-1,x) + 4*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[(9 + 16*x + 4*x^2)*P[n - 1, x] + (16*x + 8*x^2)*D[P[n - 1, x], x] + 4*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] = (9 + 16*x + 4*x^2)*v[i - 1] + (16*x + 8*x^2)*deriv(v[i - 1], x) + 4*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

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. [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.