Real-rooted atlas

A038259 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-9 + 20\,x + 20\,n\right)\,P_{n}(x) = \left(121 + 440\,x + 400\,x^{2} + 220\,n + 400\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1312051
256112404002201
385913366037200800087451
412112168728013464009920001600003306801
51625151121121003436400044880000248000003200000120981251
62109768119501812072672600013745600001346400000595200000640000004323001801
7267505711295367534013651268400339138800004810960000037699200000138880000001280000000151763129451

All 15 rows

Technical details
Node id
n-37b90a022eb32967 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-23d6a8d3aff0c2ae

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:          31,           20;
  2:         561,         1240,           400;
  3:        8591,        33660,         37200,           8000;
  4:      121121,       687280,       1346400,         992000,         160000;
  5:     1625151,     12112100,      34364000,       44880000,       24800000,        3200000;
  6:    21097681,    195018120,     726726000,     1374560000,     1346400000,      595200000,       64000000;
  7:   267505711,   2953675340,   13651268400,    33913880000,    48109600000,    37699200000,    13888000000,     1280000000;
  8:  3332306241,  42800913760,  236294027200,   728067648000,  1356555200000,  1539507200000,  1005312000000,   317440000000,   25600000000;
  9: 40942546271, 599815123380, 3852082238400, 14177641632000, 32763044160000, 48835987200000, 46185216000000, 25850880000000, 7142400000000, 512000000000;
%F Formula
The row polynomials satisfy (20*n-9+20*x)*P(n,x) = (220*n+121+(400*n+440)*x+400*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[((220*n + 121 + (400*n + 440)*x + 400*x^2)*P[n - 1, x])/(20*n - 9 + 20*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] = ((220*n + 121 + (400*n + 440)*x + 400*x^2)*v[i - 1])/(20*n - 9 + 20*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.