Real-rooted atlas

A038226 → basis A007318 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 64 + 176\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
064176121361
15122112290413316859
2409622528464644259214641130321
3327682252806195208518405856401610512476099
42621442162688743424013629440140553607730448177156147045881
5209715220185088832634881908121602623667202164525449920741619487171893871739
6167772161845493768881438722442395648419786752046176542723174637312124717894421435888116983563041
7134217728166094438491351941122930874777660449292288831177768967619129548844898441984154338394322357947691322687697779

All 15 rows

Technical details
Node id
n-b6d91d50fb3f7b0a · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e348a50f4e54dede

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:         64,          176,          121;
  1:        512,         2112,         2904,          1331;
  2:       4096,        22528,        46464,         42592,          14641;
  3:      32768,       225280,       619520,        851840,         585640,         161051;
  4:     262144,      2162688,      7434240,      13629440,       14055360,        7730448,        1771561;
  5:    2097152,     20185088,     83263488,     190812160,      262366720,      216452544,       99207416,       19487171;
  6:   16777216,    184549376,    888143872,    2442395648,     4197867520,     4617654272,     3174637312,     1247178944,      214358881;
  7:  134217728,   1660944384,   9135194112,   29308747776,    60449292288,    83117776896,    76191295488,    44898441984,    15433839432,    2357947691;
  8: 1073741824,  14763950080,  91351941120,  334957117440,   805990563840,  1329884430336,  1523825909760,  1197291786240,   617353577280,  188635815280,   25937424601;
  9: 8589934592, 129922760704, 893218979840, 3684528291840, 10132452802560, 19504971644928, 26819336011776, 26340419297280, 18109038266880, 8299975872320, 2282493364888, 285311670611;
%F Formula
The row polynomials satisfy P(n,x) = (8+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 176*x + 121*x^2.
%t Mathematica
P[0, x_] := 64 + 176*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 11*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] = 64 + 176*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (8 + 11*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

Candidate equalities ?

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.