Real-rooted atlas

A200536 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 14\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1814628
26422429216836784
35122688585667764392151221621952
44096286728755215232016513611424049248120961296614656
53276828672011264002616320397888041397442984160147168047520090720777617210368
62621442752512132218883842048075217920104520192105702976783901444231008016208640418348865318446656481890304
720971522569011214588313650898534412189614082119735296276021401627340839682070160512119235110451424934416104614434618752457228827993613492928512

All 15 rows

Technical details
Node id
n-feafdaba4f567b18 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a0b8814ff2967252

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         8,         14,           6;
  2:        64,        224,         292,         168,           36;
  3:       512,       2688,        5856,        6776,         4392,         1512,           216;
  4:      4096,      28672,       87552,      152320,       165136,       114240,         49248,         12096,          1296;
  5:     32768,     286720,     1126400,     2616320,      3978880,      4139744,       2984160,       1471680,        475200,         90720,          7776;
  6:    262144,    2752512,    13221888,    38420480,     75217920,    104520192,     105702976,      78390144,      42310080,      16208640,       4183488,       653184,        46656;
  7:   2097152,   25690112,   145883136,   508985344,   1218961408,   2119735296,    2760214016,    2734083968,    2070160512,    1192351104,     514249344,    161046144,     34618752,      4572288,      279936;
  8:  16777216,  234881024,  1539309568,  6268387328,  17752784896,  37077254144,   59071774720,   73234079744,   71399743744,   54925559808,   33227873280,  15641966592,   5617092096,   1487517696,   273964032,    31352832,    1679616;
  9: 134217728, 2113929216, 15703474176, 73106718720, 239015559168, 582767345664, 1098172465152, 1635341008896, 1950905714688, 1878405369344, 1463179286016, 919879317504, 463291508736, 184391230464, 56719512576, 13011425280, 2096160768, 211631616, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (8+14*x+6*x^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[(8 + 14*x + 6*x^2)*P[n - 1, x]];
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] = (8 + 14*x + 6*x^2)*v[i - 1]); 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

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