Real-rooted atlas

A013610 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 3\,x + 9\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(4 + 36\,x + 108\,x^{2} + 108\,x^{3} + 36\,n x + 180\,n x^{2} + 216\,n x^{3} + 36\,n^{2} x^{2} + 108\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
144852
21638412961696
3642304155522764845568
4256122881244164423685184001097728
5102461440829440442368010368000895795224641536
64096294912497664035389440124416000214990848146313216526385152
716384137625627869184247726080116121600030098718724096770048229323571210838081536
86553662914561486356481585446912928972800032105299968655483207687338354278434828517376216895848448
926214428311552764411904951268147266886041600288947699712786579849216132090377011212538266255365159780352004243427688448
10104857612582912038220595205435817984044590694400023115815976967865798492160176120502681602507653251072020639121408000749200107110481501299408896
11419430455364812818685624320298969989120280284364800016951598383104692190267310081937325529497603677891434905604540606709760003296480471285761069932053790721541515302141952
121677721624159191048969099673615945066086401681706188800011623953176985655375221384806418598325083176964413469721886720726497073561600079115531310858245135673858195456150682097575526428780816368664576
13671088641046898278442399380275282914343649289716524646400075555695650406441135878742999041611854840542003245900085107621888944446195630080001371335876054876161335275203130818567835469073927372820970668254298112531424756029718528
1426843545645097156608197863774617642210938585088544125380198400470124328491417628795115120099328128948387243360256428400794337804288105777973910568960019198702264768266242492513712510861312219393134069966438411743574222406942722888816545234944009718767995865530368
Technical details
Node id
n-be2861d5659b27e4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-10c18c8945c6e24a

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:      4,       48;
  2:     16,      384,      1296;
  3:     64,     2304,     15552,      27648;
  4:    256,    12288,    124416,     442368,      518400;
  5:   1024,    61440,    829440,    4423680,    10368000,      8957952;
  6:   4096,   294912,   4976640,   35389440,   124416000,    214990848,    146313216;
  7:  16384,  1376256,  27869184,  247726080,  1161216000,   3009871872,   4096770048,    2293235712;
  8:  65536,  6291456, 148635648, 1585446912,  9289728000,  32105299968,  65548320768,   73383542784,   34828517376;
  9: 262144, 28311552, 764411904, 9512681472, 66886041600, 288947699712, 786579849216, 1320903770112, 1253826625536, 515978035200;
%F Formula
The row polynomials satisfy (1+(9*n-3)*x+9*n^2*x^2)*P(n,x) = (4+(36*n+36)*x+(36*n^2+180*n+108)*x^2+(108*n^2+216*n+108)*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[((4 + (36*n + 36)*x + (36*n^2 + 180*n + 108)*x^2 + (108*n^2 + 216*n + 108)*x^3)*P[n - 1, x])/(1 + (9*n - 3)*x + 9*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] = ((4 + (36*n + 36)*x + (36*n^2 + 180*n + 108)*x^2 + (108*n^2 + 216*n + 108)*x^3)*v[i - 1])/(1 + (9*n - 3)*x + 9*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

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.