Real-rooted atlas

A038333 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(7 + 19\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 49 + 266\,x + 361\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
049266361676
134327937581685917576
2240126068106134192052130321456976
316807228095123823033609104561235247609911881376
4117649191599813001415470527409578593510399615847045881308915776
5823543156473171274138675763960651564503605254790587123052481698938717398031810176
65764801125178536118919609264556359282190305047047560909592645469487325005681738416983563041208827064576
74035360798578097110702764828677841772442759784359227490843260741355485923372157678974759610699644715833226876977795429503678976

All 15 rows

Technical details
Node id
n-56e0ac5356639602 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8f7225f9e9fdccc1

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:         49,         266,          361;
  1:        343,        2793,         7581,          6859;
  2:       2401,       26068,       106134,        192052,         130321;
  3:      16807,      228095,      1238230,       3360910,        4561235,         2476099;
  4:     117649,     1915998,     13001415,      47052740,       95785935,       103996158,        47045881;
  5:     823543,    15647317,    127413867,     576396065,     1564503605,      2547905871,      2305248169,       893871739;
  6:    5764801,   125178536,   1189196092,    6455635928,    21903050470,     47560909592,     64546948732,     50056817384,     16983563041;
  7:   40353607,   985780971,  10702764828,   67784177244,   275978435922,    749084326074,   1355485923372,   1576789747596,   1069964471583,    322687697779;
  8:  282475249,  7667185330,  93649192245,  677841772440,  3219748419090,  10487180565036,  23721003659010,  36791760777240,  37448756505405,  22588138844530,   6131066257801;
  9: 1977326743, 59037327041, 801220866985, 6524227059735, 35417232609990, 134585483917962, 365303456348754, 708241394961870, 961184750305395, 869643345514405, 472092101850677, 116490258898219;
%F Formula
The row polynomials satisfy P(n,x) = (7+19*x)*P(n-1,x) for n >= 1, with P(0,x) = 49 + 266*x + 361*x^2.
%t Mathematica
P[0, x_] := 49 + 266*x + 361*x^2;
P[n_, x_] := P[n, x] = Expand[(7 + 19*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] = 49 + 266*x + 361*x^2; for(n = 1, N, my(i = n + 1); v[i] = (7 + 19*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.