Real-rooted atlas

A094305 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
26242454
3106012080270
4151203604802401215
521210840168016806725103
6283361680448067205376179220412
736504302410080201602419216128460878732
8457205040201605040080640806404608011520295245
955990792036960110880221760295680253440126720281601082565
1066132011880633602217605322248870401013760760320337920675843897234
11781716171601029604118401153152230630432947203294720219648087859215974413817466
129121842402416016072072023063045381376922521611531520102502406150144223641637273648361131
13105273032760240240120120043243201153152023063040345945603843840030750720167731205591040860160167403915
14120336043680349440192192076876802306304052715520922521601230028801230028808945664044728320137625601966080573956280
Technical details
Node id
n-7933941f4c6059fa · revision 0 · JSON
Definition source
derived:reversal
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:  3,   6;
  2:  6,  24,   24;
  3: 10,  60,  120,    80;
  4: 15, 120,  360,   480,    240;
  5: 21, 210,  840,  1680,   1680,    672;
  6: 28, 336, 1680,  4480,   6720,   5376,   1792;
  7: 36, 504, 3024, 10080,  20160,  24192,  16128,   4608;
  8: 45, 720, 5040, 20160,  50400,  80640,  80640,  46080,  11520;
  9: 55, 990, 7920, 36960, 110880, 221760, 295680, 253440, 126720, 28160;
%F Formula
The row polynomials satisfy P(n,x) = (1+6*x+12*x^2+8*x^3)*P(n-3,x) - (3+12*x+12*x^2)*P(n-2,x) + (3+6*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 6*x, P(2,x) = 6 + 24*x + 24*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[2, x_] := 6 + 24*x + 24*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 6*x + 12*x^2 + 8*x^3)*P[n - 3, x] - (3 + 12*x + 12*x^2)*P[n - 2, x] + (3 + 6*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] = 3 + 6*x; v[3] = 6 + 24*x + 24*x^2; for(n = 3, N, my(i = n + 1); v[i] = (1 + 6*x + 12*x^2 + 8*x^3)*v[i - 3] - (3 + 12*x + 12*x^2)*v[i - 2] + (3 + 6*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

Candidate equalities ?

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.