Real-rooted atlas

A038328 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(18 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 144\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
032414416484
1583238888646410648
210497693312311044608256234256
3188956820995209331202073602304010245153632
4340122244534963225194240746496012441601105924096113379904
5612220032952342272634894848235146240522547206967296516096163842494357888
6110199605761959104102415237476352677221171218811699203344302083715891223592966553654875873536
719835929036839671858073635263873843218284971622460949905408135444234242006581248191102976106168322621441207269217792

All 15 rows

Technical details
Node id
n-3a5b707c177aa910 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-0205823d5b324a40

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:            324,             144,              16;
  1:           5832,            3888,             864,              64;
  2:         104976,           93312,           31104,            4608,            256;
  3:        1889568,         2099520,          933120,          207360,          23040,           1024;
  4:       34012224,        45349632,        25194240,         7464960,        1244160,         110592,          4096;
  5:      612220032,       952342272,       634894848,       235146240,       52254720,        6967296,        516096,        16384;
  6:    11019960576,     19591041024,     15237476352,      6772211712,     1881169920,      334430208,      37158912,      2359296,       65536;
  7:   198359290368,    396718580736,    352638738432,    182849716224,    60949905408,    13544423424,    2006581248,    191102976,    10616832,     262144;
  8:  3570467226624,   7934371614720,   7934371614720,   4701849845760,  1828497162240,   487599243264,   90296156160,  11466178560,   955514880,   47185920,   1048576;
  9: 64268410079232, 157100557971456, 174556175523840, 116370783682560, 51720348303360, 16090775027712, 3575727783936, 567575838720, 63063982080, 4671406080, 207618048, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = (18+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 144*x + 16*x^2.
%t Mathematica
P[0, x_] := 324 + 144*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 4*x)*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] = 324 + 144*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 4*x)*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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.