Real-rooted atlas

A272867 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 18\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1918633
281324432216361089
3729437410206116646804194421635937
4656152488174960314928332424209952777601555212961185921
559049590490255879062985609710280976276864735202799360758160116640777639135393
653144163772923401222410628820021611934030044131229225318420029420896053040314928006718464839808466561291467969
74782969669615664240899181607077584406233500472318491289334938312886585305662232922083214155168120365481631744742455847232587865627993642618442977

All 15 rows

Technical details
Node id
n-32697a227a12c599 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-3c4a6b4bc3495597

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,         18,           6;
  2:        81,        324,         432,          216,            36;
  3:       729,       4374,       10206,        11664,          6804,          1944,           216;
  4:      6561,      52488,      174960,       314928,        332424,        209952,         77760,         15552,          1296;
  5:     59049,     590490,     2558790,      6298560,       9710280,       9762768,       6473520,       2799360,        758160,        116640,          7776;
  6:    531441,    6377292,    34012224,    106288200,     216119340,     300441312,     292253184,     200294208,      96053040,      31492800,       6718464,        839808,         46656;
  7:   4782969,   66961566,   424089918,   1607077584,    4062335004,    7231849128,    9334938312,    8865853056,    6223292208,    3214155168,    1203654816,     317447424,      55847232,      5878656,       279936;
  8:  43046721,  688747536,  5050815264,  22499086176,   68032951056,  147851137728,  238561739136,  291212661888,  271604614752,  194141774592,  106027439616,   43807744512,   13438607616,   2962842624,    443418624,    40310784,    1679616;
  9: 387420489, 6973568802, 58113073350, 297538935552, 1047585002256, 2690247875616, 5216573837664, 7802132087808, 9117639881568, 8383435008192, 6078426587712, 3467614261248, 1545651507456, 531406987776, 137953580544, 26121388032, 3401222400, 272097792, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (9+18*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[(9 + 18*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] = (9 + 18*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.