Real-rooted atlas

A106246 → θ+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-1 - 4\,x - 6\,x^{2} - 4\,x^{3} - x^{4}\right)\,P_{n-4}(x) + \left(4 + 12\,x + 12\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(-6 - 12\,x - 6\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
\(P_{2}(x) = 12 + 14\,x + 3\,x^{2}\)
\(P_{3}(x) = 28 + 54\,x + 30\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
21214329
32854304116
455152144525408
5963504803008061312
61547021275116054011473952
72321274289835002380882154811328
8333214458808904805043681344200931232
9460340210944199922268016380739219442521083456
106165150190354080055860506523045011760270031011217344
11804750231350772201240801358281025645280017820363037412553984
121027105844936813750025393532630429752819245686625259604752444131386496
1312881453474880232804486200718146768768600600339768135850366086084520143416064
141590195021100193778328808801471470181080916610881135134570570205205502327644602158302592
Technical details
Node id
n-4d95a5003e2e6368 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8144d683f06ddae1

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   4,    2;
  2:  12,   14,     3;
  3:  28,   54,    30,     4;
  4:  55,  152,   144,    52,     5;
  5:  96,  350,   480,   300,    80,     6;
  6: 154,  702,  1275,  1160,   540,   114,    7;
  7: 232, 1274,  2898,  3500,  2380,   882,  154,    8;
  8: 333, 2144,  5880,  8904,  8050,  4368, 1344,  200,   9;
  9: 460, 3402, 10944, 19992, 22680, 16380, 7392, 1944, 252, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(1+4*x+6*x^2+4*x^3+x^4)*P(n-4,x) + (4+12*x+12*x^2+4*x^3)*P(n-3,x) - (6+12*x+6*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 12 + 14*x + 3*x^2, P(3,x) = 28 + 54*x + 30*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[2, x_] := 12 + 14*x + 3*x^2;
P[3, x_] := 28 + 54*x + 30*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*P[n - 4, x] + (4 + 12*x + 12*x^2 + 4*x^3)*P[n - 3, x] - (6 + 12*x + 6*x^2)*P[n - 2, x] + (4 + 4*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] = 4 + 2*x; v[3] = 12 + 14*x + 3*x^2; v[4] = 28 + 54*x + 30*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*v[i - 4] + (4 + 12*x + 12*x^2 + 4*x^3)*v[i - 3] - (6 + 12*x + 6*x^2)*v[i - 2] + (4 + 4*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.