Real-rooted atlas

A038303 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(121 + 44\,x + 4\,x^{2} + 22\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113215
2165524221
3205799015683211
4252891645639604161646137
53074612528908228013200104032656903
6370417336895321517340329120396002496649282325
7442890255185842225826724708092011519201108805824128130323843
8526153617708624400414867376137742528283236803686144295680133122561819706993
96216407549947076510663776196002489204256619841376101965248110584327603202995251225287652351
10730963784211243281509809470765106042517464000124460212802479365504339884160315955201900800665601024349948479309
1185593501183316081203252621367609660780694522774440233846052000547624936329091006848106820736086887680464640014643220484825047214715
1299859084713852054244028399219297443903144109408772862404167136646640112246104960021904997452831169166336320462208023170048011151360319488409666309934579369
13116121849938677259633620256010267051723691896167244513827248711157023605602166911056252848639978816008136141911041012997905929257797120602421248263577606922248192908625319776759
14134638577361330932514117982829563634870683584140249248275445769611707115967907363982479332191361011225159584641945599152640028476496688643151549040642592183193615334359046150144014909441638412417879370282373
Technical details
Node id
n-93504e416597f2ce · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-29cf522dddac13d2

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:         13,          2;
  2:        165,         52,          4;
  3:       2057,        990,        156,          8;
  4:      25289,      16456,       3960,        416,        16;
  5:     307461,     252890,      82280,      13200,      1040,        32;
  6:    3704173,    3689532,    1517340,     329120,     39600,      2496,       64;
  7:   44289025,   51858422,   25826724,    7080920,   1151920,    110880,     5824,    128;
  8:  526153617,  708624400,  414867376,  137742528,  28323680,   3686144,   295680,  13312,   256;
  9: 6216407549, 9470765106, 6377619600, 2489204256, 619841376, 101965248, 11058432, 760320, 29952, 512;
%F Formula
The row polynomials satisfy (2*n+9+2*x)*P(n,x) = (22*n+121+(4*n+44)*x+4*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[((22*n + 121 + (4*n + 44)*x + 4*x^2)*P[n - 1, x])/(2*n + 9 + 2*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] = ((22*n + 121 + (4*n + 44)*x + 4*x^2)*v[i - 1])/(2*n + 9 + 2*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.