Real-rooted atlas

A094305 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 48\,x + 96\,x^{2} + 64\,x^{3}\right)\,P_{n-3}(x) + \left(-12 - 48\,x - 48\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 12\,x\)
\(P_{2}(x) = 24 + 96\,x + 96\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
161218
2249696216
3804809606402160
4240192057607680384019440
5672672026880537605376021504163296
61792215041075202867204300803440641146881306368
7460864512387072129024025804803096576206438458982410077696
8115201843201290240516096012902400206438402064384011796480294912075582720
928160506880405504018923520567705601135411201513881601297612806488064014417920554273280
1067584135168012165120648806402270822405449973769083289601038090240778567680346030080692060163990767616
1115974435143683514368021086208084344832023616552964723310592674758656067475865604498391040179935641632715571228298170368
1237273689456649840230465601536029520691209446621184220421160963778648473647233105920419849830402519098982491603599361526726656198087192576
13860160223641602683699201968046080984023040035424829440944662118401889324236802833986355203148873728002519098982401374053990404580179968070464307201371372871680
141966080550502407156531205725224960314887372801259549491203778648473608636910796801511459389440201527918592020152791859201465657589760732828794880225485783040322122547209403699691520
Technical details
Node id
n-2352aed893dd9749 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-11aec07ce60b6d61

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,     12;
  2:    24,     96,      96;
  3:    80,    480,     960,      640;
  4:   240,   1920,    5760,     7680,     3840;
  5:   672,   6720,   26880,    53760,    53760,     21504;
  6:  1792,  21504,  107520,   286720,   430080,    344064,    114688;
  7:  4608,  64512,  387072,  1290240,  2580480,   3096576,   2064384,    589824;
  8: 11520, 184320, 1290240,  5160960, 12902400,  20643840,  20643840,  11796480,  2949120;
  9: 28160, 506880, 4055040, 18923520, 56770560, 113541120, 151388160, 129761280, 64880640, 14417920;
%F Formula
The row polynomials satisfy P(n,x) = (8+48*x+96*x^2+64*x^3)*P(n-3,x) - (12+48*x+48*x^2)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 12*x, P(2,x) = 24 + 96*x + 96*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 12*x;
P[2, x_] := 24 + 96*x + 96*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 48*x + 96*x^2 + 64*x^3)*P[n - 3, x] - (12 + 48*x + 48*x^2)*P[n - 2, x] + (6 + 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] = 6 + 12*x; v[3] = 24 + 96*x + 96*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 48*x + 96*x^2 + 64*x^3)*v[i - 3] - (12 + 48*x + 48*x^2)*v[i - 2] + (6 + 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.