Real-rooted atlas

A286724 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + x\right)\,P_{n-1}(x) + \left(8 + 4\,x\right)\,P_{n-1}'(x) + \left(4 + 4\,x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
31710128
4139111211272
5147314443903613344
619091218057510101055149472
7291793376206155535271402175781852928
8512930773094353494211742875779454137105116758016
91018170891579854168514511621203896273287019090471961361369082624
102250495523376009128922435464606371054769690042683554384158841170017118996922368
1154780588561977067810106367138564520220597720353077221036090961222243410828120180452101240294124544
121455367098923275213565639119381577797456783148972051332836323501570771032611510226425317257015356552667525316974172532736
134188844878585783515660940460630286006522262403218865190052337518680156968657654645903996406032493233521166145752687740380823001218457925292032
14129801943909905927159570186681012181595413070866897960841109046214182161687065317262647589033044914890732194128054692831212939723832451544841945280635117342688736329728
15430744777712089139421906867009365480096370696660955353236821453839969139963471119481148804546597306615928556804460311525415312840572463718827196030654824592117537185724714354061263513011737051136
Technical details
Node id
n-f3d7765079876101 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-dbdc887cb9bc3435

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          1;
   2:          3,          1;
   3:         17,         10,          1;
   4:        139,        111,         21,         1;
   5:       1473,       1444,        390,        36,        1;
   6:      19091,      21805,       7510,      1010,       55,       1;
   7:     291793,     376206,     155535,     27140,     2175,      78,      1;
   8:    5129307,    7309435,    3494211,    742875,    77945,    4137,    105,     1;
   9:  101817089,  157985416,   85145116,  21203896,  2732870,  190904,   7196,   136,   1;
  10: 2250495523, 3760091289, 2243546460, 637105476, 96900426, 8355438, 415884, 11700, 171, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + (8+4*x)*d/dx P(n-1,x) + (4+4*x)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + (8 + 4*x)*D[P[n - 1, x], x] + (4 + 4*x)*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (3 + x)*v[i - 1] + (8 + 4*x)*deriv(v[i - 1], x) + (4 + 4*x)*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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.