Real-rooted atlas

A038279 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(9\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2541872
3729486811296
487488748291632420736
59841513122065610145801215311040
61062882177147011809803936606561043744478976
7111602612232052218600435826686020667152755621530962705664
8114791256267846264267846264148803480496011609920232110224852488859963392
91162261467309936391236159245642410616376100442349026784626444641044425152817714711609505792
1011622614670348678440104649045868036159245640180796228206026540940133923132019131876015943230590490154793410560
111150638852333835462841105753194261655113950454802983137765301193255106123314597517063135190807891898855845851019486172043273019392
12112971814592441422998683886903833113980690383311398046025554093202147859191016715953063672170465015160284108358603156759540210450636637729226748301344768
131101475192275944059007691036807748474335668974983048174067312372861305358999321926961396108474160439888813547448310169489051231136220601231136220674614316420726199347727917481984
14106758364789818462619580755878925239161511756113084786406992494237322005827056542393203496225129952534887283766508449624209416271124063878079094890517077212652470070193322611501074669615664493714625921024
Technical details
Node id
n-579b67e8fdaf1fdc · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f9762973ec1089d7

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          0;
  1:          3;
  2:         54,         18;
  3:        729,        486,         81;
  4:       8748,       8748,       2916,        324;
  5:      98415,     131220,      65610,      14580,       1215;
  6:    1062882,    1771470,    1180980,     393660,      65610,      4374;
  7:   11160261,   22320522,   18600435,    8266860,    2066715,    275562,    15309;
  8:  114791256,  267846264,  267846264,  148803480,   49601160,   9920232,  1102248,   52488;
  9: 1162261467, 3099363912, 3615924564, 2410616376, 1004423490, 267846264, 44641044, 4251528, 177147;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (9*n+3*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((9*n + 3*n*x)*P[n - 1, x])/(n - 1)]];
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; for(n = 2, N, my(i = n + 1); v[i] = ((9*n + 3*n*x)*v[i - 1])/(n - 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.