Real-rooted atlas

A038243 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 2\,n x\right)\,P_{n}(x) = \left(18 + 24\,x + 8\,x^{2} + 12\,n x + 8\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16814
2369648180
32168648642562200
412966912103686144128026000
577765184010368092160384006144300000
6466563732489331201105920691200221184286723400000
7279936261273678382081161216096768004644864120422413107238000000
81679616179159046270566411147673611612160074317824289013766291456589824420000000
9100776961209323524837294081003290624125411328010032906245202247681698693123185049626214404600000000
1060466176806215680362797056085996339201254113280012039487488780337152033973862409555148801572864001153433650000000000
11362797056532102348826605117440709469798401182449664001324343623681030045040645605687296021021327360519045120076126617650331648540000000000
122176782336348285173761915568455685675758387201064204697600136218201292812360540487688072189706243783838924801245708288002740558233636238786562181038085800000000000
1313060694016226385362944135831217766444270915420169223107379200132812746260481377317368627210493846618112590278872268824291311616007125451407361413312675841701209702493952409662000000000000
14783641640961462797729792950818524364833806880866304774741019852801239585631764481446183237058561259261594173448263904211763240809403514880149634479554563957275492352714508075008789200240644026531840660000000000000
Technical details
Node id
n-f046fc914a7585a0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-925da9e442f58149

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:        6,         8;
  2:       36,        96,        48;
  3:      216,       864,       864,        256;
  4:     1296,      6912,     10368,       6144,       1280;
  5:     7776,     51840,    103680,      92160,      38400,       6144;
  6:    46656,    373248,    933120,    1105920,     691200,     221184,     28672;
  7:   279936,   2612736,   7838208,   11612160,    9676800,    4644864,   1204224,    131072;
  8:  1679616,  17915904,  62705664,  111476736,  116121600,   74317824,  28901376,   6291456,   589824;
  9: 10077696, 120932352, 483729408, 1003290624, 1254113280, 1003290624, 520224768, 169869312, 31850496, 2621440;
%F Formula
The row polynomials satisfy (3+2*n*x)*P(n,x) = (18+(12*n+24)*x+(8*n+8)*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[((18 + (12*n + 24)*x + (8*n + 8)*x^2)*P[n - 1, x])/(3 + 2*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] = ((18 + (12*n + 24)*x + (8*n + 8)*x^2)*v[i - 1])/(3 + 2*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.