Real-rooted atlas

A038322 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(35 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1351651
2122511202562601
34287558800268804096132651
4150062527440001881600573440655366765201
55252187512005000010976000050176000114688001048576345025251
6183826562550421000005762400000351232000012042240002202009601677721617596287801
76433929687520588575000028235760000021512960000098344960000269746176004110417920268435456897410677851
82251875390625823543000000013176688000000120472576000006884147200000251763097600057545850880075161927680429496729645767944570401
97881563867187532427005625000059295096000000063248102400000043370127360000019826343936000060423143424000118380036096001352914698240687194767362334165173090451
102758547353515625126105021875000002594160450000000031624051200000000252992409600000001387844075520000052870250496000001381100421120000236760072192000240518168576001099511627776119042423827613001
1196549157373046875485504334218750000110972419250000000015219074640000000001391458252800000000890533281792000000407100928819200000132930915532800000303842092646400004629974745088000423311976693760175921860444166071163615208263051
123379220508056640625185374382156250000004660841608500000000071022348320000000000730515582720000000005343199690752000000028497065017344000000111661969047552000003190341972787200000648196464312320000888955151056896007388718138654720281474976710656309629344375621415601
131182727177819824218757028778656757812500001927893574425000000000323151684856000000000036931621126400000000003038944824115200000000185230922612736000000084676993194393600000029032111952363520000073732347815526400000134824864576962560001680933376543948800128071114403348480450359962737049615791096563156692195651
144139545122369384765625264930887831640625000007872232095568750000000014394938689040000000000018096494351936000000000016545366264627200000000011345394010030080000000059273895236075520000000237095580944302080000007225770085921587200000165160459106779136000027455245150217830400031377423028820377600220676381741154304072057594037927936805345924720991301978201
Technical details
Node id
n-5515fc4124201f2c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b04825ff48559760

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 0)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:              1;
  1:             35,              16;
  2:           1225,            1120,             256;
  3:          42875,           58800,           26880,            4096;
  4:        1500625,         2744000,         1881600,          573440,           65536;
  5:       52521875,       120050000,       109760000,        50176000,        11468800,         1048576;
  6:     1838265625,      5042100000,      5762400000,      3512320000,      1204224000,       220200960,       16777216;
  7:    64339296875,    205885750000,    282357600000,    215129600000,     98344960000,     26974617600,     4110417920,      268435456;
  8:  2251875390625,   8235430000000,  13176688000000,  12047257600000,   6884147200000,   2517630976000,   575458508800,    75161927680,    4294967296;
  9: 78815638671875, 324270056250000, 592950960000000, 632481024000000, 433701273600000, 198263439360000, 60423143424000, 11838003609600, 1352914698240, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (35+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(35 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (35 + 16*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.