Real-rooted atlas

A272867 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 12\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1412622
2169619214436484
364576201634563024129621610648
425630721536041472656646220834560103681296234256
510241536099840368640852480128563212787208294403369607776077765153632
640967372858982427648008432640175841282565734426376192189734409331200298598455987246656113379904
71638434406432686081857945670447104188116992364234752518897664546352128423263232237758976940584962482099239191042799362494357888

All 15 rows

Technical details
Node id
n-ec0bc4323ea11152 · 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:      4,      12,        6;
  2:     16,      96,      192,       144,         36;
  3:     64,     576,     2016,      3456,       3024,        1296,         216;
  4:    256,    3072,    15360,     41472,      65664,       62208,       34560,       10368,         1296;
  5:   1024,   15360,    99840,    368640,     852480,     1285632,     1278720,      829440,       336960,        77760,         7776;
  6:   4096,   73728,   589824,   2764800,    8432640,    17584128,    25657344,    26376192,     18973440,      9331200,      2985984,       559872,        46656;
  7:  16384,  344064,  3268608,  18579456,   70447104,   188116992,   364234752,   518897664,    546352128,    423263232,    237758976,     94058496,     24820992,     3919104,      279936;
  8:  65536, 1572864, 17301504, 115605504,  524353536,  1709309952,  4137025536,  7575109632,  10597588992,  11362664448,   9308307456,   5768921088,   2654539776,   877879296,   197074944,   26873856,    1679616;
  9: 262144, 7077888, 88473600, 679477248, 3588489216, 13823115264, 40205942784, 90200604672, 158113824768, 218072383488, 237170737152, 202951360512, 135695056896, 69979521024, 27250089984, 7739670528, 1511654400, 181398528, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (4+12*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[(4 + 12*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] = (4 + 12*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.