Real-rooted atlas

A094305 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-6 + 11\,n - 6\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(16\,n + 24\,n x + 12\,n x^{2} + 2\,n x^{3} - 24\,n^{2} - 36\,n^{2} x - 18\,n^{2} x^{2} - 3\,n^{2} x^{3} + 8\,n^{3} + 12\,n^{3} x + 6\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-3}(x) + \left(-36\,n - 36\,n x - 9\,n x^{2} + 48\,n^{2} + 48\,n^{2} x + 12\,n^{2} x^{2} - 12\,n^{3} - 12\,n^{3} x - 3\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(36\,n + 18\,n x - 30\,n^{2} - 15\,n^{2} x + 6\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)
\(P_{2}(x) = 24 + 12\,x\)
\(P_{3}(x) = 120 + 120\,x + 30\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2241236
312012030270
4480720360601620
51680336025208401058505
6537613440134406720168016840824
716128483846048040320151203024252183708
846080161280241920201600100800302405040360787320
91267205068808870408870405544002217605544079204953247695
1033792015206403041280354816026611201330560443520950401188066012990780
11878592439296098841601317888011531520691891228828808236801544401716085850664042
12223641612300288307507204612608046126080322882561614412857657601441440240240240241092193444524
13559104033546240922521601537536001729728001383782408072064034594560108108002402400360360327601365725416965
141376256089456640268369920492011520615014400553512960369008640184504320691891201921920038438405241604368016802678462640
Technical details
Node id
n-fde1ec7f640113ad · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-11aec07ce60b6d61

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:      0;
  1:      3;
  2:     24,     12;
  3:    120,    120,     30;
  4:    480,    720,    360,     60;
  5:   1680,   3360,   2520,    840,    105;
  6:   5376,  13440,  13440,   6720,   1680,    168;
  7:  16128,  48384,  60480,  40320,  15120,   3024,   252;
  8:  46080, 161280, 241920, 201600, 100800,  30240,  5040,  360;
  9: 126720, 506880, 887040, 887040, 554400, 221760, 55440, 7920, 495;
%F Formula
The row polynomials satisfy (n^3-6*n^2+11*n-6)*P(n,x) = (8*n^3-24*n^2+16*n+(12*n^3-36*n^2+24*n)*x+(6*n^3-18*n^2+12*n)*x^2+(n^3-3*n^2+2*n)*x^3)*P(n-3,x) - (12*n^3-48*n^2+36*n+(12*n^3-48*n^2+36*n)*x+(3*n^3-12*n^2+9*n)*x^2)*P(n-2,x) + (6*n^3-30*n^2+36*n+(3*n^3-15*n^2+18*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 24 + 12*x, P(3,x) = 120 + 120*x + 30*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[2, x_] := 24 + 12*x;
P[3, x_] := 120 + 120*x + 30*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 24*n^2 + 16*n + (12*n^3 - 36*n^2 + 24*n)*x + (6*n^3 - 18*n^2 + 12*n)*x^2 + (n^3 - 3*n^2 + 2*n)*x^3)*P[n - 3, x] - (12*n^3 - 48*n^2 + 36*n + (12*n^3 - 48*n^2 + 36*n)*x + (3*n^3 - 12*n^2 + 9*n)*x^2)*P[n - 2, x] + (6*n^3 - 30*n^2 + 36*n + (3*n^3 - 15*n^2 + 18*n)*x)*P[n - 1, x])/(n^3 - 6*n^2 + 11*n - 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; v[3] = 24 + 12*x; v[4] = 120 + 120*x + 30*x^2; for(n = 4, N, my(i = n + 1); v[i] = ((8*n^3 - 24*n^2 + 16*n + (12*n^3 - 36*n^2 + 24*n)*x + (6*n^3 - 18*n^2 + 12*n)*x^2 + (n^3 - 3*n^2 + 2*n)*x^3)*v[i - 3] - (12*n^3 - 48*n^2 + 36*n + (12*n^3 - 48*n^2 + 36*n)*x + (3*n^3 - 12*n^2 + 9*n)*x^2)*v[i - 2] + (6*n^3 - 30*n^2 + 36*n + (3*n^3 - 15*n^2 + 18*n)*x)*v[i - 1])/(n^3 - 6*n^2 + 11*n - 6)); 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.