Real-rooted atlas

A008277 → (θ+1)² → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 14\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 18\,x + 16\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(3 + 10\,x + 11\,x^{2} + 4\,x^{3}\right)\,P_{n-2}''(x) + \left(5 + 4\,x\right)\,P_{n-1}(x) + \left(-2 - 6\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2549
32230961
49918410216401
54711090855260252701
623866544654029005553618961
7128674054448804281408015105049139469
87368126110436552625765696950191521820641075953
94466201752414278561723099161075935284004410342952818698573
10285645712263164217529402067082011441535377521273458080760454510073580113
111921724389427338174697479186654380119384815472030021156455317212801485006710121650025421
1213561044867886372814454488401711650776123960573057043958416857178831755768372190525828095701445985936561
1310011599015357751048123305649481600150895212920949470677462557223438795385352409927968589275289504288571326016957353337549
147714225057439011176281084612412521528559410561359683925607992121737631672729088850765458415410164771854052201439838468468417927196570744041489
156190458551037296056978298347610634914940648086001449881692350943338371976421114545852130135554120278270699854084009930404527123262462201056965237302255889219168717
Technical details
Node id
n-b27306dd6cb81497 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6c4ea92fd5c5ccfc

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:       5,        4;
   3:      22,       30,        9;
   4:      99,      184,      102,       16;
   5:     471,     1090,      855,      260,       25;
   6:    2386,     6544,     6540,     2900,      555,      36;
   7:   12867,    40544,    48804,    28140,     8015,    1050,     49;
   8:   73681,   261104,   365526,   257656,    96950,   19152,   1820,    64;
   9:  446620,  1752414,  2785617,  2309916,  1075935,  284004,  41034,  2952,   81;
  10: 2856457, 12263164, 21752940, 20670820, 11441535, 3775212, 734580, 80760, 4545, 100;
%F Formula
The row polynomials satisfy P(n,x) = (5+14*x+9*x^2)*P(n-2,x) + (6+18*x+16*x^2+4*x^3)*d/dx P(n-2,x) + (3+10*x+11*x^2+4*x^3)*d^2/dx^2 P(n-2,x) + (5+4*x)*P(n-1,x) - (2+6*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[(5 + 14*x + 9*x^2)*P[n - 2, x] + (6 + 18*x + 16*x^2 + 4*x^3)*D[P[n - 2, x], x] + (3 + 10*x + 11*x^2 + 4*x^3)*D[P[n - 2, x], {x, 2}] + (5 + 4*x)*P[n - 1, x] - (2 + 6*x + 4*x^2)*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; v[2] = 5 + 4*x; for(n = 3, N, my(i = n); v[i] = (5 + 14*x + 9*x^2)*v[i - 2] + (6 + 18*x + 16*x^2 + 4*x^3)*deriv(v[i - 2], x) + (3 + 10*x + 11*x^2 + 4*x^3)*deriv(deriv(v[i - 2], x), x) + (5 + 4*x)*v[i - 1] - (2 + 6*x + 4*x^2)*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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.