Real-rooted atlas

A086645 → Veronese 2 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(191 - 381\,x + 189\,x^{2} + x^{3}\right)\,P_{n-6}(x) + \left(-64 + 192\,x - 192\,x^{2} + 64\,x^{3}\right)\,P_{n-6}'(x) + \left(3342 - 2252\,x - 66\,x^{2}\right)\,P_{n-5}(x) + \left(-128 + 1264\,x - 1120\,x^{2} - 16\,x^{3}\right)\,P_{n-5}'(x) + \left(9041 + 146\,x - 3\,x^{2}\right)\,P_{n-4}(x) + \left(1152 + 5184\,x + 64\,x^{2}\right)\,P_{n-4}'(x) + \left(1860 + 52\,x\right)\,P_{n-3}(x) + \left(-2048 + 416\,x + 32\,x^{2}\right)\,P_{n-3}'(x) + \left(-111 + 3\,x\right)\,P_{n-2}(x) + \left(1472 + 128\,x\right)\,P_{n-2}'(x) + \left(14\right)\,P_{n-1}(x) + \left(-384 - 16\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 1 + 2\,x\)
\(P_{4}(x) = 1 + 30\,x\)
\(P_{5}(x) = 1 + 140\,x + 3\,x^{2}\)
\(P_{6}(x) = 1 + 420\,x + 135\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
3123
413031
511403144
61420135556
71990148542480
812002900936411376
913640386107280549536
101612013127474256765212416
1119690377910503880242256915712
12114630959310258658437306513863934976
13121252220641310816624367735563756716785408
14129900468682538630800265586751381380227571289856
151409509324315121687020152108775186486301433258301953024
Technical details
Node id
n-254cbc520517186c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-090aca1ee1bd5ff6

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1;
   3: 1,    2;
   4: 1,   30;
   5: 1,  140,      3;
   6: 1,  420,    135;
   7: 1,  990,   1485,     4;
   8: 1, 2002,   9009,   364;
   9: 1, 3640,  38610,  7280,   5;
  10: 1, 6120, 131274, 74256, 765;
%F Formula
The row polynomials satisfy P(n,x) = (191-381*x+189*x^2+x^3)*P(n-6,x) - (64-192*x+192*x^2-64*x^3)*d/dx P(n-6,x) + (3342-2252*x-66*x^2)*P(n-5,x) - (128-1264*x+1120*x^2+16*x^3)*d/dx P(n-5,x) + (9041+146*x-3*x^2)*P(n-4,x) + (1152+5184*x+64*x^2)*d/dx P(n-4,x) + (1860+52*x)*P(n-3,x) - (2048-416*x-32*x^2)*d/dx P(n-3,x) - (111-3*x)*P(n-2,x) + (1472+128*x)*d/dx P(n-2,x) + 14*P(n-1,x) - (384+16*x)*d/dx P(n-1,x) for n >= 7, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 1 + 2*x, P(4,x) = 1 + 30*x, P(5,x) = 1 + 140*x + 3*x^2, P(6,x) = 1 + 420*x + 135*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 1 + 2*x;
P[4, x_] := 1 + 30*x;
P[5, x_] := 1 + 140*x + 3*x^2;
P[6, x_] := 1 + 420*x + 135*x^2;
P[n_, x_] := P[n, x] = Expand[(191 - 381*x + 189*x^2 + x^3)*P[n - 6, x] - (64 - 192*x + 192*x^2 - 64*x^3)*D[P[n - 6, x], x] + (3342 - 2252*x - 66*x^2)*P[n - 5, x] - (128 - 1264*x + 1120*x^2 + 16*x^3)*D[P[n - 5, x], x] + (9041 + 146*x - 3*x^2)*P[n - 4, x] + (1152 + 5184*x + 64*x^2)*D[P[n - 4, x], x] + (1860 + 52*x)*P[n - 3, x] - (2048 - 416*x - 32*x^2)*D[P[n - 3, x], x] - (111 - 3*x)*P[n - 2, x] + (1472 + 128*x)*D[P[n - 2, x], x] + 14*P[n - 1, x] - (384 + 16*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 1 + 2*x; v[4] = 1 + 30*x; v[5] = 1 + 140*x + 3*x^2; v[6] = 1 + 420*x + 135*x^2; for(n = 7, N, my(i = n); v[i] = (191 - 381*x + 189*x^2 + x^3)*v[i - 6] - (64 - 192*x + 192*x^2 - 64*x^3)*deriv(v[i - 6], x) + (3342 - 2252*x - 66*x^2)*v[i - 5] - (128 - 1264*x + 1120*x^2 + 16*x^3)*deriv(v[i - 5], x) + (9041 + 146*x - 3*x^2)*v[i - 4] + (1152 + 5184*x + 64*x^2)*deriv(v[i - 4], x) + (1860 + 52*x)*v[i - 3] - (2048 - 416*x - 32*x^2)*deriv(v[i - 3], x) - (111 - 3*x)*v[i - 2] + (1472 + 128*x)*deriv(v[i - 2], x) + 14*v[i - 1] - (384 + 16*x)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

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  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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [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
  5. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  8. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023