Real-rooted atlas

A094305 → x ↦ x+1 → basis A013610

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 81\,x + 81\,x^{2} + 27\,x^{3}\right)\,P_{n-3}(x) + \left(-27 - 54\,x - 27\,x^{2}\right)\,P_{n-2}(x) + \left(9 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 9\,x\)
\(P_{2}(x) = 54 + 108\,x + 54\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19918
25410854216
32708108102702160
41215486072904860121519440
55103255155103051030255155103163296
620412122472306180408240306180122472204121306368
77873255112416533722755620275562016533725511247873210077696
8295245236196082668601653372020667150165337208266860236196029524575582720
9108256597430853897234090935460136403190136403190909354603897234097430851082565554273280
103897234389723401753755304676680808184191409821029688184191404676680801753755303897234038972343990767616
11138174661519921267599606302279881890455976378063836692926383669292455976378022798818907599606301519921261381746628298170368
1248361131580333572319183464610639448820239387598453830201575244685685044383020157522393875984510639448820319183464658033357248361131198087192576
1316740391521762508951305750537047877519690119693799225215448838605287265118140287265118140215448838605119693799225478775196901305750537021762508951674039151371372871680
14573956280803538792052230021480208920085920574530236280114906047256017235907088401969817952960172359070884011490604725605745302362802089200859205223002148080353879205739562809403699691520
Technical details
Node id
n-32c54683b478c3e5 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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,       9;
  2:      54,     108,       54;
  3:     270,     810,      810,      270;
  4:    1215,    4860,     7290,     4860,      1215;
  5:    5103,   25515,    51030,    51030,     25515,      5103;
  6:   20412,  122472,   306180,   408240,    306180,    122472,    20412;
  7:   78732,  551124,  1653372,  2755620,   2755620,   1653372,   551124,    78732;
  8:  295245, 2361960,  8266860, 16533720,  20667150,  16533720,  8266860,  2361960,  295245;
  9: 1082565, 9743085, 38972340, 90935460, 136403190, 136403190, 90935460, 38972340, 9743085, 1082565;
%F Formula
The row polynomials satisfy P(n,x) = (27+81*x+81*x^2+27*x^3)*P(n-3,x) - (27+54*x+27*x^2)*P(n-2,x) + (9+9*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 9 + 9*x, P(2,x) = 54 + 108*x + 54*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 9*x;
P[2, x_] := 54 + 108*x + 54*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 81*x + 81*x^2 + 27*x^3)*P[n - 3, x] - (27 + 54*x + 27*x^2)*P[n - 2, x] + (9 + 9*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 + 9*x; v[3] = 54 + 108*x + 54*x^2; for(n = 3, N, my(i = n + 1); v[i] = (27 + 81*x + 81*x^2 + 27*x^3)*v[i - 3] - (27 + 54*x + 27*x^2)*v[i - 2] + (9 + 9*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.