Real-rooted atlas

A075503 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + x\right)\,P_{n-1}(x) + \left(8 + 8\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 89 + 26\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
089261116
110095475111608
21345711124184684126512
32101052321015929046501251506272
43747753510446218259512167409815174110904896
574565473119738823559549979292451626115183892311260196480
61628999841300540541617416325723826415603569872615382643161229616795948288
73870424189780587013065557296171561551116328419078198861125910383356724509163691192555854336
89930342815932300357338498184609536078162864824188098235420130758590350030677560266898807792545015874270090240
927340167242321696366348085556354029916314325735020786981495973660721048466992052625446746094746605002124108051144555391191722734405632
108031545836493292227807745598044227578461277467410715075565642362484710185036312993000652549619769443279967477036274416608345272172478016251463616658708921028608
112505085321762831375088440063210861848386044225065584558660657151065412456019576232947180993466599674314680297353493269433113160841975754673753433542827362444942243027411244984405847416832
12826165199464341705265897200505231787832901321083421305071987789247721812444628680795836931201108189360939750090105665295911885716107630389923922161353551800347859199588266790413695807292430221185419511128108852396032
1328707262835597618369986708030188278337431326192166589290792098900496274994373959132091276314625995285643715290226173701974468427782107258951649393398846243222300975551011560913810493965107813709503512341127253898938953988259751388329293165540442112
141047731789671001235265382801838069853605934455502703254907045641524096703829475104334832166223340774007902703638325661386086259348851776522768398264200424416747309872574421831489990665011872432793731240801579540311503442283856252496218214640281350703205169201104116624460755616909033472
Technical details
Node id
n-5f4b4a174c534dfa · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b122d3773b1f4b51

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             89,             26,              1;
  1:           1009,            547,             51,              1;
  2:          13457,          11124,           1846,             84,             1;
  3:         210105,         232101,          59290,           4650,           125,            1;
  4:        3747753,        5104462,        1825951,         216740,          9815,          174,           1;
  5:       74565473,      119738823,       55954997,        9292451,        626115,        18389,         231,         1;
  6:     1628999841,     3005405416,     1741632572,      382641560,      35698726,      1538264,       31612,       296,        1;
  7:    38704241897,    80587013065,    55729617156,    15511163284,    1907819886,    112591038,     3356724,     50916,      369,      1;
  8:   993034281593,  2300357338498,  1846095360781,   628648241880,   98235420130,   7585903500,   306775602,   6689880,    77925,    450,   1;
  9: 27340167242321, 69636634808555, 63540299163143, 25735020786981, 4959736607210, 484669920526, 25446746094, 746605002, 12410805, 114455, 539, 1;
%F Formula
The row polynomials satisfy P(n,x) = (9+x)*P(n-1,x) + (8+8*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 89 + 26*x + x^2.
%t Mathematica
P[0, x_] := 89 + 26*x + x^2;
P[n_, x_] := P[n, x] = Expand[(9 + x)*P[n - 1, x] + (8 + 8*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 89 + 26*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (9 + x)*v[i - 1] + (8 + 8*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 - 1, v[i])))};
T(9)

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.