Real-rooted atlas

A013620 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(25 + 30\,x + 9\,x^{2} + 15\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18311
255489112
3350495216271088
42125420029708648110240
512500318753150014850324024394208
6718752250002868751890006682511664729851968
74062501509375236250020081259922502806654082421877602176
82265625975000018112500189000001204875047628001122660139968656167108864
912500000611718751316250001630125001275750006506325021432600433026047239219683587202560
10683593753750000009175781251316250000122259375076545000032531625091854000162384751574640590495100273664
1137109375022558593756187500000100933593751085906250080691187504209975000153363375037889775059541075519631217714744023414784
122001953125133593750004060546875074250000000908402343757818525000048414712500216513000006901351875151559100021434787017006112531441377957122048
1310742187500780761718752605078125005278710937507239375000007085538281255082041250002697391125001055500875002990585812559108049007599606305526986415943233229815406592
14573730468754511718750001639599609375364710937500055426464843756081075000000495987679687530492247500001416130340625492567075000125604604125225685278002659862205178564176478296927487790694400
Technical details
Node id
n-1131bd1f361ee0e9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-37bd9a5ad19991c8

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:        8,        3;
  2:       55,       48,         9;
  3:      350,      495,       216,        27;
  4:     2125,     4200,      2970,       864,        81;
  5:    12500,    31875,     31500,     14850,      3240,      243;
  6:    71875,   225000,    286875,    189000,     66825,    11664,      729;
  7:   406250,  1509375,   2362500,   2008125,    992250,   280665,    40824,    2187;
  8:  2265625,  9750000,  18112500,  18900000,  12048750,  4762800,  1122660,  139968,   6561;
  9: 12500000, 61171875, 131625000, 163012500, 127575000, 65063250, 21432600, 4330260, 472392, 19683;
%F Formula
The row polynomials satisfy (3*n+2+3*x)*P(n,x) = (15*n+25+(9*n+30)*x+9*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[((15*n + 25 + (9*n + 30)*x + 9*x^2)*P[n - 1, x])/(3*n + 2 + 3*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] = ((15*n + 25 + (9*n + 30)*x + 9*x^2)*v[i - 1])/(3*n + 2 + 3*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.