Real-rooted atlas

A008277 → θ+1 → D → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 - 3\,x + x^{3}\right)\,P_{n-3}(x) + \left(-3 - 6\,x - 3\,x^{2}\right)\,P_{n-3}'(x) + \left(-1 - 3\,x - 3\,x^{2} - x^{3}\right)\,P_{n-3}''(x) + \left(-5 - 6\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(2 + 2\,x\right)\,P_{n-2}'(x) + \left(1 + 2\,x + x^{2}\right)\,P_{n-2}''(x) + \left(6 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)
\(P_{3}(x) = 12 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
222
312618
4626012134
532045018020970
6171231201860420307142
79604213361680057408404254362
8566741477561438926580014700151256430390
935179210488301212372693420209160330122520723551178
102293862767544010239780701778026693105720406720039609030539462
111568221658062906876031206985022032072040871655413952401267205940110273515066
1211217960845450451276369207269339468037286238012300472825066272310860022473085801322548046294
13837905016368194101668076534246924637720426515232016564427884113629526511190464350003789501201215624657033258
1465221658343085678596062176062312699545246404852523676021690933264636664028012314942641555854301253252061261216380182247492590438
15528074019082673457659665824574574007174548654005530631106002796626392569484146672021745306440336360024034667633023183160955500218402102573112450970
Technical details
Node id
n-51062c3aa8b3c844 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d9103e4f061eeed7

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       0;
   2:       2;
   3:      12,       6;
   4:      62,      60,       12;
   5:     320,     450,      180,      20;
   6:    1712,    3120,     1860,     420,      30;
   7:    9604,   21336,    16800,    5740,     840,     42;
   8:   56674,  147756,   143892,   65800,   14700,   1512,    56;
   9:  351792, 1048830,  1212372,  693420,  209160,  33012,  2520,   72;
  10: 2293862, 7675440, 10239780, 7017780, 2669310, 572040, 67200, 3960, 90;
%F Formula
The row polynomials satisfy P(n,x) = -(2+3*x-x^3)*P(n-3,x) - (3+6*x+3*x^2)*d/dx P(n-3,x) - (1+3*x+3*x^2+x^3)*d^2/dx^2 P(n-3,x) - (5+6*x+3*x^2)*P(n-2,x) + (2+2*x)*d/dx P(n-2,x) + (1+2*x+x^2)*d^2/dx^2 P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 2, P(3,x) = 12 + 6*x.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[3, x_] := 12 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(2 + 3*x - x^3)*P[n - 3, x] - (3 + 6*x + 3*x^2)*D[P[n - 3, x], x] - (1 + 3*x + 3*x^2 + x^3)*D[P[n - 3, x], {x, 2}] - (5 + 6*x + 3*x^2)*P[n - 2, x] + (2 + 2*x)*D[P[n - 2, x], x] + (1 + 2*x + x^2)*D[P[n - 2, x], {x, 2}] + (6 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; v[3] = 12 + 6*x; for(n = 4, N, my(i = n); v[i] = -(2 + 3*x - x^3)*v[i - 3] - (3 + 6*x + 3*x^2)*deriv(v[i - 3], x) - (1 + 3*x + 3*x^2 + x^3)*deriv(deriv(v[i - 3], x), x) - (5 + 6*x + 3*x^2)*v[i - 2] + (2 + 2*x)*deriv(v[i - 2], x) + (1 + 2*x + x^2)*deriv(deriv(v[i - 2], x), x) + (6 + 3*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, 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.