Real-rooted atlas

A071951 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + x\right)\,P_{n-1}(x) + \left(4 + 4\,x\right)\,P_{n-1}'(x) + \left(1 + 2\,x + x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
31310124
48195231200
566910284184412160
66955131457530138275129088
7885051976221451033957237671181474688
81346209345052730788931129155163459892517519177344
9239985216922154472537364335080086737134561976190122481206583808
104939564671580045745189724507610569304122766480823250952616734923726033915339046400
11115965425334066280006654930338605358088975281163379727418166959481320906744442360682894501156585673728
1230730150507311702535914231753776309891130860299661350942109012210103391877498472328704662489701073774911845958315160713750528
139110471500693373960103434046148887062018251631961219124234581089029355700629615224715147390948454381725841464759219240150681962627401189548126736384
14299893197116059131854075866144923574157044999662198097004643666114232553585198933056596644572751635132744932424478840201218307163733341736908075031340631275492317702905384321024
151088867403499390551007248989607790984271794588328591008217935650110525899320058514690519872294356477042375943684562284338816750760586421660099951115657454669170109480626979970391648202711341344180323946168320
Technical details
Node id
n-7c4933365ce8d2f7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-cb8f91bdd594c2ea

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:        13,         10,          1;
   4:        81,         95,         23,          1;
   5:       669,       1028,        418,         44,         1;
   6:      6955,      13145,       7530,       1382,        75,        1;
   7:     88505,     197622,     145103,      39572,      3767,      118,       1;
   8:   1346209,    3450527,    3078893,    1129155,    163459,     8925,     175,     1;
   9:  23998521,   69221544,   72537364,   33508008,   6737134,   561976,   19012,   248,   1;
  10: 493956467, 1580045745, 1897245076, 1056930412, 276648082, 32509526, 1673492, 37260, 339, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + (4+4*x)*d/dx P(n-1,x) + (1+2*x+x^2)*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] + (4 + 4*x)*D[P[n - 1, x], x] + (1 + 2*x + x^2)*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] + (4 + 4*x)*deriv(v[i - 1], x) + (1 + 2*x + x^2)*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.