Real-rooted atlas

A094305 → x ↦ x+1 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 108\,x + 144\,x^{2} + 64\,x^{3}\right)\,P_{n-3}(x) + \left(-27 - 72\,x - 48\,x^{2}\right)\,P_{n-2}(x) + \left(9 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 12\,x\)
\(P_{2}(x) = 54 + 144\,x + 96\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
191221
25414496294
3270108014406403430
4121564801296011520384036015
5510334020907201209608064021504352947
6204121632965443209676809676805160961146883294172
7787327348322939328653184087091206967296309657658982429647548
829524531492801469664039191040653184006967296046448640176947202949120259416045
91082565129907806928416021555072043110144057480192051093504029196288097320960144179202219448385
103897234519631203117787201108546560258660864041385738244598415360350355456017517772805190451206920601618643366434
1113817466202656168135104112054041644801441110528026900729856358676398083415965696022773104640101213798402699034624327155712154231485954
124836113177377809656743727042521943424075658302720161404379136251073478656286941118464239117598720141699317760566797271041374053990415267266561259557135291
13167403915290166786023213342880113487454080378291513600907899632640161404379136021520583884802152058388480159411732480085019590656030916214784068702699520704643072010173346092735
14573956280107138505609285337152049521798144018157992652804842131374080968426274816014756971806720172164671078401530352631808010202350878720494659436544016488647884803382286745603221225472081386768741880
Technical details
Node id
n-1eff416cdfba16f3 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-c91d5d506c107735

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:       9,       12;
  2:      54,      144,       96;
  3:     270,     1080,     1440,       640;
  4:    1215,     6480,    12960,     11520,      3840;
  5:    5103,    34020,    90720,    120960,     80640,     21504;
  6:   20412,   163296,   544320,    967680,    967680,    516096,    114688;
  7:   78732,   734832,  2939328,   6531840,   8709120,   6967296,   3096576,    589824;
  8:  295245,  3149280, 14696640,  39191040,  65318400,  69672960,  46448640,  17694720,  2949120;
  9: 1082565, 12990780, 69284160, 215550720, 431101440, 574801920, 510935040, 291962880, 97320960, 14417920;
%F Formula
The row polynomials satisfy P(n,x) = (27+108*x+144*x^2+64*x^3)*P(n-3,x) - (27+72*x+48*x^2)*P(n-2,x) + (9+12*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 9 + 12*x, P(2,x) = 54 + 144*x + 96*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 12*x;
P[2, x_] := 54 + 144*x + 96*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 108*x + 144*x^2 + 64*x^3)*P[n - 3, x] - (27 + 72*x + 48*x^2)*P[n - 2, x] + (9 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 12*x; v[3] = 54 + 144*x + 96*x^2; for(n = 3, N, my(i = n + 1); v[i] = (27 + 108*x + 144*x^2 + 64*x^3)*v[i - 3] - (27 + 72*x + 48*x^2)*v[i - 2] + (9 + 12*x)*v[i - 1]); 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.