Real-rooted atlas

A038255 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 4\,n x\right)\,P_{n}(x) = \left(49 + 56\,x + 16\,x^{2} + 28\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17815
24911248209
3343117610082562783
4240110976141127168128035937
51680796040164640125440448006144453871
611764980673617287201756160940800258048286725636785
782354365883441694145621512960153664006322176140492813107269090879

All 15 rows

Technical details
Node id
n-14677aff3512ef46 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-604d756e1453b62f

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:        7,         8;
  2:       49,       112,         48;
  3:      343,      1176,       1008,        256;
  4:     2401,     10976,      14112,       7168,       1280;
  5:    16807,     96040,     164640,     125440,      44800,       6144;
  6:   117649,    806736,    1728720,    1756160,     940800,     258048,     28672;
  7:   823543,   6588344,   16941456,   21512960,   15366400,    6322176,   1404928,    131072;
  8:  5764801,  52706752,  158120256,  240945152,  215129600,  118013952,  39337984,   7340032,   589824;
  9: 40353607, 415065672, 1423082304, 2529924096, 2710632960, 1858719744, 826097664, 231211008, 37158912, 2621440;
%F Formula
The row polynomials satisfy (7+4*n*x)*P(n,x) = (49+(28*n+56)*x+(16*n+16)*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[((49 + (28*n + 56)*x + (16*n + 16)*x^2)*P[n - 1, x])/(7 + 4*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] = ((49 + (28*n + 56)*x + (16*n + 16)*x^2)*v[i - 1])/(7 + 4*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.