Real-rooted atlas

A038326 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(12 + 23\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1122335
21445525291225
317289936190441216742875
4207361589764570565840162798411500625
5248832238464091411201752048016790460643634352521875
62985984343388161645401604204915206044565604634166961480358891838265625
73583180848074342427642746888830321920169247836801946350123212435014676340482544764339296875
8429981696659305267244228395008169542180864406194808320622832039424596880704448326863242912783109852812251875390625
9515978035289006211072682380951552305175925555287738078597121681646506444821487705360128176506151172488457586410348180115266146378815638671875
106191736422411867494809601023571427328052315872952320175476157194240403595161546752644631160803840706024604689920507455184620880216138319375560414265112136492758547353515625
11743008370688156650931486721501238093414408632119037132803308978964234240887909355402854417018262645221376232988119547673602232802812331872014265129078786960546829948020166895280975791392796549157373046875
12891610044825620507031030988821617828545167361381139045941248059561621356216320182655638825730048408438303485313024671005784297299968803809012439473920684726195781774080393717562574520096137204605139605488219146244320203213379220508056640625
1310699320537907226659140340285443065801139132825621545769116683468810324014368410828803561784957101735936910233933481554739217446150391729799168250788411881115863042670432163548918912020473313253875044992107019592008892280643418681411395170076504036361936467383118272717781982421875
141283918464548864344518121320611844292121594785955843290626556002566144173443441389301923846648665253256573747219114912603112649523241870760940151518003270220755326712441651289726520695243675443285987915666275188966459930971524979677158428716923855719428638484678108805326520344115928363245387498094139545122369384765625
Technical details
Node id
n-454026b53ecaf199 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-4b5617cf7459dd0c

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:         12,          23;
  2:        144,         552,          529;
  3:       1728,        9936,        19044,         12167;
  4:      20736,      158976,       457056,        584016,        279841;
  5:     248832,     2384640,      9141120,      17520480,      16790460,        6436343;
  6:    2985984,    34338816,    164540160,     420491520,     604456560,      463416696,      148035889;
  7:   35831808,   480743424,   2764274688,    8830321920,   16924783680,    19463501232,    12435014676,     3404825447;
  8:  429981696,  6593052672,  44228395008,  169542180864,  406194808320,   622832039424,   596880704448,   326863242912,   78310985281;
  9: 5159780352, 89006211072, 682380951552, 3051759255552, 8773807859712, 16816465064448, 21487705360128, 17650615117248, 8457586410348, 1801152661463;
%F Formula
The row polynomials satisfy P(n,x) = (12+23*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[(12 + 23*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] = (12 + 23*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

Same fingerprint ?

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.