Real-rooted atlas

A038243 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + n x\right)\,P_{n}(x) = \left(25 + 10\,x + x^{2} + 5\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
22520348
3125150454324
462510004508052160
53125625037501000125614256
6156253750028125100001875180793312
778125218750196875875002187531502458606528
83906251250000131250070000021875042000490032093919104
9195312570312508437500525000019687504725007350072004051025194240
1097656253906250052734375375000001640625047250009187501200001012550011161243136
11488281252148437503222656252578125001289062504331250010106250165000018562513750605121027924992
1224414062511718750001933593750171875000096679687537125000010106250019800000278437527500018150720136530347008
1312207031256347656250114257812501117187500069824218753016406250938437500214500000361968754468750393250234008451441358864384
1461035156253417968750066650390625710937500004887695312523460937500821132812521450000004222968756256250068818755460002957598015261213880320
Technical details
Node id
n-f482e744be1cefd4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6ae5c48b72ea3401

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:       5,       2;
  2:      25,      20,       3;
  3:     125,     150,      45,       4;
  4:     625,    1000,     450,      80,       5;
  5:    3125,    6250,    3750,    1000,     125,      6;
  6:   15625,   37500,   28125,   10000,    1875,    180,     7;
  7:   78125,  218750,  196875,   87500,   21875,   3150,   245,    8;
  8:  390625, 1250000, 1312500,  700000,  218750,  42000,  4900,  320,   9;
  9: 1953125, 7031250, 8437500, 5250000, 1968750, 472500, 73500, 7200, 405, 10;
%F Formula
The row polynomials satisfy (5+n*x)*P(n,x) = (25+(5*n+10)*x+(n+1)*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[((25 + (5*n + 10)*x + (n + 1)*x^2)*P[n - 1, x])/(5 + n*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] = ((25 + (5*n + 10)*x + (n + 1)*x^2)*v[i - 1])/(5 + n*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.