Real-rooted atlas

A038323 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 9\,x + 9\,n\right)\,P_{n}(x) = \left(121 + 198\,x + 81\,x^{2} + 99\,n + 81\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
120929
231936081760
345988613486072918800
462557165528155034583206561448000
58198962815065372438023255106561005904910400000
610468315442743847600675567038840313943857085880531441236800000
71310955146595038451394643096159614185510558617303955692517440174047829695312000000
81617435193943887700823742138420334714343042873055339015204408912474683101276527504043046721117760000000
9197210170521310122506333822745188246410377373406777965446564654349649182052595203125492761599677484097803874204892585600000000
10238152716791177489153468058955512784851146823556472014423349090150122003378038086981524473770263905097544061793567995577484097800348678440156320000000000
112853116706110235771189623098785713096666019455319219000528383883022682028558231198497020130557376283298738703271890326582558210706797292479505767092568220313810596091218560000000000
1233952088802709308136604259880127316442396468631628567147997605252936189130135613091873289931251404816157294603105857423769408133297249417051539189906985284073410758778654753145430616028242953648126214400000000000
1340171883222028839723943899169531802599134920298049653412534622754925135589078929801229187068256451591195529152915365848591947843433526045423164822627592173286424242166954585219117278228780823525191138734316794850602541865828329561152000000000000
14472961156371854750616572859756288250260846564768039757091636666525160156408249484061675123313416844789030962581292843338548339215195247524765851213532317853407803456359243075167862882183408945451303575252160079791424881986470145069497117224319321202287679245496111960320000000000000
Technical details
Node id
n-6afaaae13a3aced3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-83cfe7e8f5f2fb1e

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:          20,            9;
  2:         319,          360,           81;
  3:        4598,         8613,         4860,          729;
  4:       62557,       165528,       155034,        58320,         6561;
  5:      819896,      2815065,      3724380,      2325510,       656100,        59049;
  6:    10468315,     44274384,     76006755,     67038840,     31394385,      7085880,       531441;
  7:   131095514,    659503845,   1394643096,   1596141855,   1055861730,    395569251,     74401740,     4782969;
  8:  1617435193,   9438877008,  23742138420,  33471434304,  28730553390,  15204408912,   4746831012,   765275040,   43046721;
  9: 19721017052, 131012250633, 382274518824, 641037737340, 677796544656, 465434964918, 205259520312, 54927615996, 7748409780, 387420489;
%F Formula
The row polynomials satisfy (9*n+2+9*x)*P(n,x) = (99*n+121+(81*n+198)*x+81*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[((99*n + 121 + (81*n + 198)*x + 81*x^2)*P[n - 1, x])/(9*n + 2 + 9*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] = ((99*n + 121 + (81*n + 198)*x + 81*x^2)*v[i - 1])/(9*n + 2 + 9*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

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.