Real-rooted atlas

A038291 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(8 + x + n\right)\,P_{n}(x) = \left(81 + 18\,x + x^{2} + 9\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110111
299201120
39722973011300
49477388859440114000
591854473859720990501150000
68857355511241421551944014856011600000
785030566200145192893433169534020207970117000000
88131047368024448248005805143824663390544322772801180000000
9774840978731794257306110016744017401157360411941028164835649011900000000
1073609892917748409780365897128510203667201860043502314720819901701166404455100120000000000
1169735688020809708822014261625379013416228045280600848040920957042436548312741016038054451101210000000000
12659002251789836828256240485825293206170465015160402486841356734420352818419140727483684691115213840653412012200000000000
13621344980258285670292732575439383665560210524293722655401129927010464657875114591244096151992126011821609867760552779927722130123000000000000
145846291405156786988297236148599692049127992538379043928073683502802911551231637956244175350419291824881922659862205183891708948647735380890091401240000000000000
Technical details
Node id
n-dfd02436707ba66e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a35ab9f2b26f7628

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:        10,         1;
  2:        99,        20,         1;
  3:       972,       297,        30,        1;
  4:      9477,      3888,       594,       40,        1;
  5:     91854,     47385,      9720,      990,       50,       1;
  6:    885735,    551124,    142155,    19440,     1485,      60,     1;
  7:   8503056,   6200145,   1928934,   331695,    34020,    2079,    70,    1;
  8:  81310473,  68024448,  24800580,  5143824,   663390,   54432,  2772,   80,  1;
  9: 774840978, 731794257, 306110016, 74401740, 11573604, 1194102, 81648, 3564, 90, 1;
%F Formula
The row polynomials satisfy (n+8+x)*P(n,x) = (9*n+81+(n+18)*x+x^2)*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[Together[((9*n + 81 + (n + 18)*x + x^2)*P[n - 1, x])/(n + 8 + 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] = ((9*n + 81 + (n + 18)*x + x^2)*v[i - 1])/(n + 8 + 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. [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.