Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 36\,x + 36\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19363681
2816481944259212966561
372987484374011664017496013996846656531441
465611049767348322939328734832011757312117573126718464167961643046721
55904911809801062882056687040198404640476171136793618560906992640680244480302330880604661763486784401
6531441127545841403004249353361604209012720134688407043142729497653875362816673442035205986151424035916908544130606940162176782336282429536481
74782969133923132174100071613928005728766040315043064161260169192483780482101139149824367699351219249026580162564902658016256356556946636817827847331845485491486727836416409622876792454961

All 15 rows

Technical details
Node id
n-59c900e21016c37e · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-90334921cba1a1bb

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:         9,          36,           36;
  2:        81,         648,         1944,          2592,           1296;
  3:       729,        8748,        43740,        116640,         174960,          139968,           46656;
  4:      6561,      104976,       734832,       2939328,        7348320,        11757312,        11757312,          6718464,          1679616;
  5:     59049,     1180980,     10628820,      56687040,      198404640,       476171136,       793618560,        906992640,        680244480,        302330880,          60466176;
  6:    531441,    12754584,    140300424,     935336160,     4209012720,     13468840704,     31427294976,      53875362816,      67344203520,      59861514240,       35916908544,       13060694016,        2176782336;
  7:   4782969,   133923132,   1741000716,   13928005728,    76604031504,    306416126016,    919248378048,    2101139149824,    3676993512192,    4902658016256,     4902658016256,     3565569466368,     1782784733184,      548549148672,       78364164096;
  8:  43046721,  1377495072,  20662426080,  192849310080,  1253520515520,   6016898474496,  22061961073152,   63034174494720,  141826892613120,  252136697978880,   352991377170432,   385081502367744,   320901251973120,   197477693521920,    84633297223680,    22568879259648,    2821109907456;
  9: 387420489, 13947137604, 237101339268, 2529080952192, 18968107141440, 106221399992064, 460292733298944, 1578146514167808, 4339902913961472, 9644228697692160, 17359611655845888, 25250344226684928, 29458734931132416, 27192678397968384, 19423341712834560, 10359115580178432, 3884668342566912, 914039610015744, 101559956668416;
%F Formula
The row polynomials satisfy P(n,x) = (9+36*x+36*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[(9 + 36*x + 36*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] = (9 + 36*x + 36*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.