Real-rooted atlas

A027466 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + n x\right)\,P_{n}(x) = \left(49 + 14\,x + x^{2} + 7\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1729
24928380
3343294634704
42401274488211256144
51680724010102901960175653248
61176492016841080452744036752527458752
78235431647086105884133614060025617434383932160
8576480113176688988251637647688403501152489604448933554432
94035360710376641888942644395300641058841018151562016841411256710285212672
1028247524980707214077824813539530064012353145025412184352947032928019845700112415919104
1119773267436214455478665834515538047686601358845950326123028543538386338640509355269508471220401094656
12138412872014745584183255930099302355111741601426788247539134763367609537321064891521069645575460035574100813171798691840
13968890104073598734672264626944578623231516848561442641450254451579332298923985161615085472194675481171671501079078458641183141443109011456
1467822307284927128922913963778671405873287898773780814137886212454847275272841211818818212261119660831796995233364761402643741114982245796713721512094627905536
Technical details
Node id
n-af45d624a27988ee · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-86b557643ca68de2

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,         2;
  2:       49,        28,        3;
  3:      343,       294,       63,        4;
  4:     2401,      2744,      882,      112,        5;
  5:    16807,     24010,    10290,     1960,      175,       6;
  6:   117649,    201684,   108045,    27440,     3675,     252,      7;
  7:   823543,   1647086,  1058841,   336140,    60025,    6174,    343,     8;
  8:  5764801,  13176688,  9882516,  3764768,   840350,  115248,   9604,   448,   9;
  9: 40353607, 103766418, 88942644, 39530064, 10588410, 1815156, 201684, 14112, 567, 10;
%F Formula
The row polynomials satisfy (7+n*x)*P(n,x) = (49+(7*n+14)*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[((49 + (7*n + 14)*x + (n + 1)*x^2)*P[n - 1, x])/(7 + 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 + (7*n + 14)*x + (n + 1)*x^2)*v[i - 1])/(7 + 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.