Real-rooted atlas

A013620 → basis A008288 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-100 + 60\,x - 9\,x^{2}\right)\,P_{n-4}(x) + \left(140 + 18\,x - 18\,x^{2}\right)\,P_{n-3}(x) + \left(-69 - 36\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(14 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)
\(P_{2}(x) = 39 + 18\,x\)
\(P_{3}(x) = 333 + 378\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2391857
333337881792
42391469823493249762
5156574545038475119881215112785
6969393794584715012439725467543741250919
757825328772824827843362782813304252318221530913488762
8335963120397546437352214415245223285475656537493384952488142482036
9191411371376373063626048434670411403288701251322566383017403936216721771471481524047
1010742289989449029028120385254453123716398621209521279028866877568251314824401364031959049015214660485
11595705173564444318620702795427392000280844306540162529107049814124231157193345637608549687141501916501948617154686004044
123271488471347849678341462243833813239282952364252950096753518561292301894065762336705620186415446686959222319485018122138163772921559680532406
1317822273817210253504842998475078987254320650104439081422639253858019887150253498600309511279245522323402361455216838169445087512389476441064922072619915616863976701
14964355632591250487429282662975484614119141960404420338558012367753908795210271030585402316665165656844046806260246194131164204935425029244259350933678479052225915569166961566155444221042131
Technical details
Node id
n-af2f6bf610715ee0 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-91ad29f9b5defc65

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:        0;
  1:        3;
  2:       39,        18;
  3:      333,       378,        81;
  4:     2391,      4698,      2349,       324;
  5:    15657,     45450,     38475,     11988,      1215;
  6:    96939,    379458,    471501,    243972,     54675,      4374;
  7:   578253,   2877282,   4827843,   3627828,   1330425,    231822,    15309;
  8:  3359631,  20397546,  43735221,  44152452,  23285475,   6565374,   933849,   52488;
  9: 19141137, 137637306, 362604843, 467041140, 328870125, 132256638, 30174039, 3621672, 177147;
%F Formula
The row polynomials satisfy P(n,x) = -(100-60*x+9*x^2)*P(n-4,x) + (140+18*x-18*x^2)*P(n-3,x) - (69+36*x+9*x^2)*P(n-2,x) + (14+6*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 39 + 18*x, P(3,x) = 333 + 378*x + 81*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[2, x_] := 39 + 18*x;
P[3, x_] := 333 + 378*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[-(100 - 60*x + 9*x^2)*P[n - 4, x] + (140 + 18*x - 18*x^2)*P[n - 3, x] - (69 + 36*x + 9*x^2)*P[n - 2, x] + (14 + 6*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] = 0; v[2] = 3; v[3] = 39 + 18*x; v[4] = 333 + 378*x + 81*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(100 - 60*x + 9*x^2)*v[i - 4] + (140 + 18*x - 18*x^2)*v[i - 3] - (69 + 36*x + 9*x^2)*v[i - 2] + (14 + 6*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.