Real-rooted atlas

A038277 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(324 - 198\,x + 594\,n x + 121\,n^{2} x^{2}\right)\,P_{n}(x) = \left(5832 + 10692\,x + 6534\,x^{2} + 1331\,x^{3} + 10692\,n x + 10890\,n x^{2} + 2662\,n x^{3} + 2178\,n^{2} x^{2} + 1331\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1184462
2324158410892997
35832427685880621296128702
41049761026432211701615333123660255147761
51889568230947206351048068999040329422505797836196233894
634012224498845952171478296024839654401778881500626166288868064897223460853
76122200321047576499243212530592782449113607471302300039448476144109376176141247178944258891722678
8110199605762155014512641037100734208225345344716826896688280001893526854912787508468208179593767936173630693619084736581633
919835929036843639043880962400147413452860843243073536871452700272007668783762393642525457283232145470952028162812817236482235794769100313361253029294
10357046722662487278087761920540033168026880156454053617664026143581008160002760762154461696191364557774544087282571216896025315355128338042443058438000313842837672110655748842482261
1164268410079232172810613768601611880729696591360387223782703718407394898628022400091105151097235968757803648787194244320487275236352016708134384703080420186278536200062140881859075841084880567984358007348392495182
12115683138142617633933720521834496256623761446373376929337078488924160199662262956604800028112446624289955842728093135633899264186645050290210406490223925677396632030253412054606400067112152407801864887433420268454453039439566584911904752580695787697
1320822964865671168661707550175772672545908728895012454421746487636640825344519121883687172480008222890637604812083291196256248333203968727915696131820584964222479721702162377617698246051944744000523474788780854539210382971017140916481241122885858086666766451580210476392343993045568048982
143748133675820810241282694635725343948811464083306795261542449819226222122618060813081871468916746496002302409378529347383296287268207182249592499226204965060745541058561773441483114908198592891991601018015097600329789116931938359696872169565439836984321563814836181189191617051457982130399528544371255622922512829677853813512895701
Technical details
Node id
n-538a574cb67dc127 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-141b63b59857ebd8

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:           18,            44;
  2:          324,          1584,           1089;
  3:         5832,         42768,          58806,          21296;
  4:       104976,       1026432,        2117016,        1533312,         366025;
  5:      1889568,      23094720,       63510480,       68999040,       32942250,        5797836;
  6:     34012224,     498845952,     1714782960,     2483965440,     1778881500,      626166288,       86806489;
  7:    612220032,   10475764992,    43212530592,    78244911360,    74713023000,    39448476144,    10937617614,     1247178944;
  8:  11019960576,  215501451264,  1037100734208,  2253453447168,  2689668828000,  1893526854912,   787508468208,   179593767936,   17363069361;
  9: 198359290368, 4363904388096, 24001474134528, 60843243073536, 87145270027200, 76687837623936, 42525457283232, 14547095202816, 2812817236482, 235794769100;
%F Formula
The row polynomials satisfy (324+(594*n-198)*x+121*n^2*x^2)*P(n,x) = (5832+(10692*n+10692)*x+(2178*n^2+10890*n+6534)*x^2+(1331*n^2+2662*n+1331)*x^3)*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[((5832 + (10692*n + 10692)*x + (2178*n^2 + 10890*n + 6534)*x^2 + (1331*n^2 + 2662*n + 1331)*x^3)*P[n - 1, x])/(324 + (594*n - 198)*x + 121*n^2*x^2)]];
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] = ((5832 + (10692*n + 10692)*x + (2178*n^2 + 10890*n + 6534)*x^2 + (1331*n^2 + 2662*n + 1331)*x^3)*v[i - 1])/(324 + (594*n - 198)*x + 121*n^2*x^2)); 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.