Real-rooted atlas

A013613 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 24\,n x\right)\,P_{n}(x) = \left(49 + 336\,x + 576\,x^{2} + 168\,n x + 576\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
174855
24967217282449
33437056362885529698983
4240165856508032154828816588803783457
5168075762405927040270950405806080047775744139451671
61176494840416622339203793305601219276800200658124813377208325010101425
782354339530064609892416464679936019914854400491612405766554832076836691771392176613232519

All 15 rows

Technical details
Node id
n-aa02a12fdeba1c08 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7fb21d131bc4d9e0

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,         48;
  2:       49,        672,        1728;
  3:      343,       7056,       36288,        55296;
  4:     2401,      65856,      508032,      1548288,       1658880;
  5:    16807,     576240,     5927040,     27095040,      58060800,       47775744;
  6:   117649,    4840416,    62233920,    379330560,    1219276800,     2006581248,     1337720832;
  7:   823543,   39530064,   609892416,   4646799360,   19914854400,    49161240576,    65548320768,    36691771392;
  8:  5764801,  316240512,  5692329216,  52044152832,  278807961600,   917676490752,  1835352981504,  2054739197952,   990677827584;
  9: 40353607, 2490394032, 51230962944, 546463604736, 3512980316160, 14453404729344, 38542412611584, 64724284735488, 62412703137792, 26418075402240;
%F Formula
The row polynomials satisfy (7+24*n*x)*P(n,x) = (49+(168*n+336)*x+(576*n+576)*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 + (168*n + 336)*x + (576*n + 576)*x^2)*P[n - 1, x])/(7 + 24*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 + (168*n + 336)*x + (576*n + 576)*x^2)*v[i - 1])/(7 + 24*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.