Real-rooted atlas

A143494 → θ+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-5 - 4\,x - x^{2}\right)\,P_{n-2}(x) + \left(-2 - 3\,x - x^{2}\right)\,P_{n-2}'(x) + \left(5 + 2\,x\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2426
31716336
477104394224
53726483637651464
6191540623015960130610088
7104812610824048103002140204773288
8608141734901903021024242887542423018560456
93729391196192151901798378035136570392771442494501832
102409837856868012340713933349640412401031058154518131285761037893256
11163607866377582010255707388620220451108401402506026854593124802119576011333469704
1211639320549293899887430975584890744449667645518244624242719061636345659152532780979123061379912
13865549453395316191276576796048245896824545010070023118960486418696901171885441396068310604004891712361329258414024
14671306515632859105762689558313608148602321660017556170288865073649294359074201713626429494822528725840181595770824153414290555146760
1554190360453282774216080638525885091820927472456666412959030358879280760131470110772331530730965772241332690589900559849292990988999181876152992855266696
Technical details
Node id
n-53b59547a77aa9a8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-97b1acff54b6a7ed

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:       4,       2;
   3:      17,      16,        3;
   4:      77,     104,       39,       4;
   5:     372,     648,      363,      76,       5;
   6:    1915,    4062,     3015,     960,     130,       6;
   7:   10481,   26108,    24048,   10300,    2140,     204,      7;
   8:   60814,  173490,   190302,  102424,   28875,    4242,    301,     8;
   9:  372939, 1196192,  1519017,  983780,  351365,   70392,   7714,   424,   9;
  10: 2409837, 8568680, 12340713, 9333496, 4041240, 1031058, 154518, 13128, 576, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(5+4*x+x^2)*P(n-2,x) - (2+3*x+x^2)*d/dx P(n-2,x) + (5+2*x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(5 + 4*x + x^2)*P[n - 2, x] - (2 + 3*x + x^2)*D[P[n - 2, x], x] + (5 + 2*x)*P[n - 1, x] + (1 + 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; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = -(5 + 4*x + x^2)*v[i - 2] - (2 + 3*x + x^2)*deriv(v[i - 2], x) + (5 + 2*x)*v[i - 1] + (1 + 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. [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.