Real-rooted atlas

A075504 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(10 + x\right)\,P_{n-1}(x) + \left(9 + 9\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 109 + 29\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0109291139
113516785712087
21961215259230194137267
33334513509518222558151401772583
464930698481570281025630242012300195118099811
51412648202182275079533407814442596876995230792591470169335
633766957636003600857327753178366128195055822921216032639718332113377133651
7877993653431764406841401156291034312975458908933263509861767856234723194640264141413131666247
8246595981069055214962257614217429485611133629753522019072208748013304612727483103677942690098055505113745702386627
9743530641387491832881188411471596895552738096052643343803110708221132015948563386023448171427551178327337175083151441006051489559969332887
102393123710957813643123931704910462888093693346362784663553233496595789957881756652993330720503891095144574132095637036262568126730675865204699714118562951088873011
1181812390963020066237785239382914237257703817595723304130769799322717204331294716040365224397372414651502322651447458054137231256994263498862459955435607892512164512826338321745934821229574119
12295819106407642879392384006559614100411098427671377653125762888473702815746441852607251767714806292408132572604048259116219710307961349631433227792428540821102688491360447571058523365982113304380926959131642036155726406051
131127275165444169782993762047843753207821984865970276393068006253103654904778325261011046671469173636378801933428212969660043020357787190119258881275202105085989084932446185895085121078133212465191578862775319576723803127174775502845109511412064602473269988151
1445131182248220568227721601936491718303425131122380789801177389404425157381374643442012300425995096944512906887931127798471487732065948215765789340850158846611699890211524092292541788126542205543161229772847271703145950916442624063386397980346421801181911740606519001240166094408080334073918995
Technical details
Node id
n-65ab378cd6773b3c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e1b101719f9aa420

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:            109,              29,               1;
  1:           1351,             678,              57,              1;
  2:          19612,           15259,            2301,             94,              1;
  3:         333451,          350951,           82225,           5815,            140,            1;
  4:        6493069,         8481570,         2810256,         302420,          12300,          195,           1;
  5:      141264820,       218227507,        95334078,       14442596,         876995,        23079,         259,          1;
  6:     3376695763,      6003600857,      3277531783,      661281950,       55822921,      2160326,       39718,        332,        1;
  7:    87799365343,    176440684140,    115629103431,    29754589089,     3326350986,    176785623,     4723194,      64026,      414,      1;
  8:  2465959810690,   5521496225761,   4217429485611,  1336297535220,   190722087480,  13304612727,   483103677,    9426900,    98055,    505,   1;
  9: 74353064138749, 183288118841147, 159689555273809, 60526433438031, 10708221132015, 948563386023, 44817142755, 1178327337, 17508315, 144100, 605, 1;
%F Formula
The row polynomials satisfy P(n,x) = (10+x)*P(n-1,x) + (9+9*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 109 + 29*x + x^2.
%t Mathematica
P[0, x_] := 109 + 29*x + x^2;
P[n_, x_] := P[n, x] = Expand[(10 + x)*P[n - 1, x] + (9 + 9*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] = 109 + 29*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (10 + x)*v[i - 1] + (9 + 9*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.