Real-rooted atlas

A111577 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(3 + 3\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
377115
4354815199
5214365177261783
61523310120154754017155
71234929118234967720105057173791
8112052298915286566121016232752037771843939
91120849332602336805871897042477421592903598100110564911
10122197673983448649844097302771019470286155011513335059221261143335251
11143942992510456779711081171496535640186112830382225414341477272580922515512090975391
121819256321696271391310667926907840632717136754533759140426131307762611083221751661513750187132587859331
13245266543811006403873281679630737081472429662887357331361621568150494375704439039231601224635457920975197672221539689482639
1435097447074615355070332692769531673584267051546626815007916143225078586710341049476745461337156363410577042195193674215608452757326019454609397043
155308470041299246456996785954772639961835350154701174396312994781322861202344716372828876149982684378785393844227416536226106950871027936912535260374923011174528679573503
Technical details
Node id
n-bc1a4b6b869cb25b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d43998b81d687a16

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:        2,        1;
   3:        7,        7,        1;
   4:       35,       48,       15,        1;
   5:      214,      365,      177,       26,       1;
   6:     1523,     3101,     2015,      475,      40,       1;
   7:    12349,    29118,    23496,     7720,    1050,      57,      1;
   8:   112052,   298915,   286566,   121016,   23275,    2037,     77,    1;
   9:  1120849,  3326023,  3680587,  1897042,  477421,   59290,   3598,  100,   1;
  10: 12219767, 39834486, 49844097, 30277101, 9470286, 1550115, 133350, 5922, 126, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (3+3*x)*d/dx 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[(2 + x)*P[n - 1, x] + (3 + 3*x)*D[P[n - 1, x], x]];
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] = (2 + x)*v[i - 1] + (3 + 3*x)*deriv(v[i - 1], 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. [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.