Real-rooted atlas

A135065 → x ↦ x+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(3 + 2\,x + 6\,n + 4\,n x + 3\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112820
28110836225
34328645761282000
4202554005400240040015625
5874829160388802592086401152112500
635721142884238140211680105840282243136765625
7139968653184130636814515209676803870728601681925000000
85314412834352661348888179847348320391910413063682488322073631640625
91968300118098003149280048988800489888003265920014515200414720069120051200195312500
10714492947632860142898580254041920296382240237105792131725440501811201254528018585601239041181640625
112550916818706723262355744012471148801662819840155196518410346434564926873601642291203649536048660482949127031250000
128981352971850823226345301845854511520878176728093672184327285614336416320819217346700805139763201027952641246003269222441259765625
1331248730827082233361083289334426480405952441340099205296081190447076277248313841848321569209241658118860801549836288281788416313098241605632239257812500
1410761680251004423490043525017900116066714400212788976400283718635200283718635200216166579200126097171200560431872001868106240045287424007547904007741440036864001373291015625
Technical details
Node id
n-bc1ce0156476df16 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-f06e51c139cfae05

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:      12,        8;
  2:      81,      108,       36;
  3:     432,      864,      576,      128;
  4:    2025,     5400,     5400,     2400,      400;
  5:    8748,    29160,    38880,    25920,     8640,     1152;
  6:   35721,   142884,   238140,   211680,   105840,    28224,     3136;
  7:  139968,   653184,  1306368,  1451520,   967680,   387072,    86016,    8192;
  8:  531441,  2834352,  6613488,  8817984,  7348320,  3919104,  1306368,  248832,  20736;
  9: 1968300, 11809800, 31492800, 48988800, 48988800, 32659200, 14515200, 4147200, 691200, 51200;
%F Formula
The row polynomials satisfy n^2*P(n,x) = (3*n^2+6*n+3+(2*n^2+4*n+2)*x)*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[((3*n^2 + 6*n + 3 + (2*n^2 + 4*n + 2)*x)*P[n - 1, x])/(n^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] = ((3*n^2 + 6*n + 3 + (2*n^2 + 4*n + 2)*x)*v[i - 1])/(n^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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.