Real-rooted atlas

A038303 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 + n x\right)\,P_{n}(x) = \left(100 + 20\,x + x^{2} + 10\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110212
2100403143
310006009041694
41000080001800160519965
51000001000003000040002506234256
61000000120000045000080000750036072737867
710000000140000006300000140000017500012600490831888098
810000000016000000084000000224000003500000336000196006409370256249
910000000001800000000108000000033600000063000000756000058800028800810104287177620
1010000000000200000000001350000000048000000001050000000151200000147000009600004050010001149516901511
111000000000002200000000001650000000006600000000016500000000277200000032340000026400000148500055000121012570623341222
12100000000000024000000000001980000000000880000000000247500000000475200000006468000000633600000445500002200000726001440136562168424053
131000000000000026000000000000234000000000001144000000000035750000000007722000000001201200000001372800000011583000007150000031460009360016901475322281041304
14100000000000000280000000000000273000000000000145600000000000500500000000001201200000000021021000000002745600000002702700000020020000001101100004368000118300196015863067803598275
Technical details
Node id
n-402f15d17052c967 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ae3bbdd3fbe0e44f

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:         10,          2;
  2:        100,         40,          3;
  3:       1000,        600,         90,         4;
  4:      10000,       8000,       1800,       160,        5;
  5:     100000,     100000,      30000,      4000,      250,       6;
  6:    1000000,    1200000,     450000,     80000,     7500,     360,      7;
  7:   10000000,   14000000,    6300000,   1400000,   175000,   12600,    490,     8;
  8:  100000000,  160000000,   84000000,  22400000,  3500000,  336000,  19600,   640,   9;
  9: 1000000000, 1800000000, 1080000000, 336000000, 63000000, 7560000, 588000, 28800, 810, 10;
%F Formula
The row polynomials satisfy (10+n*x)*P(n,x) = (100+(10*n+20)*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[((100 + (10*n + 20)*x + (n + 1)*x^2)*P[n - 1, x])/(10 + 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] = ((100 + (10*n + 20)*x + (n + 1)*x^2)*v[i - 1])/(10 + 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

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.