Real-rooted atlas

A060920 → Veronese 2 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 - 9\,x - 3\,x^{2} + 5\,x^{3}\right)\,P_{n-6}(x) + \left(-4 + 12\,x - 12\,x^{2} + 4\,x^{3}\right)\,P_{n-6}'(x) + \left(31 - 22\,x + 11\,x^{2} - 4\,x^{3}\right)\,P_{n-5}(x) + \left(28 - 44\,x + 20\,x^{2} - 4\,x^{3}\right)\,P_{n-5}'(x) + \left(-122 + 85\,x - 19\,x^{2}\right)\,P_{n-4}(x) + \left(-68 + 84\,x - 16\,x^{2}\right)\,P_{n-4}'(x) + \left(143 - 78\,x + 8\,x^{2}\right)\,P_{n-3}(x) + \left(68 - 76\,x + 8\,x^{2}\right)\,P_{n-3}'(x) + \left(-74 + 31\,x\right)\,P_{n-2}(x) + \left(-28 + 28\,x\right)\,P_{n-2}'(x) + \left(15 - 4\,x\right)\,P_{n-1}(x) + \left(4 - 4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
\(P_{2}(x) = 6 + x\)
\(P_{3}(x) = 22 + 9\,x\)
\(P_{4}(x) = 85 + 53\,x + x^{2}\)
\(P_{5}(x) = 322 + 273\,x + 20\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
2617
322931
485531139
532227320615
61175132219312691
74132611214003511679
814070270438755494150363
94662811469149856479454216023
101510554675692648453784510401922355
114800981839111132839226181112958773922447
1215014057006580633805016479501259501931116621867
1346308382595052028928600963471010410402993910470215751
14141137349375076512691735052977480766278535070132831295776099
Technical details
Node id
n-5b1df28f327520e1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-45f6f58dc95d25d6

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 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:     2;
  2:     6,      1;
  3:    22,      9;
  4:    85,     53,     1;
  5:   322,    273,    20;
  6:  1175,   1322,   193,    1;
  7:  4132,   6112,  1400,   35;
  8: 14070,  27043,  8755,  494,  1;
  9: 46628, 114691, 49856, 4794, 54;
%F Formula
The row polynomials satisfy P(n,x) = (7-9*x-3*x^2+5*x^3)*P(n-6,x) - (4-12*x+12*x^2-4*x^3)*d/dx P(n-6,x) + (31-22*x+11*x^2-4*x^3)*P(n-5,x) + (28-44*x+20*x^2-4*x^3)*d/dx P(n-5,x) - (122-85*x+19*x^2)*P(n-4,x) - (68-84*x+16*x^2)*d/dx P(n-4,x) + (143-78*x+8*x^2)*P(n-3,x) + (68-76*x+8*x^2)*d/dx P(n-3,x) - (74-31*x)*P(n-2,x) - (28-28*x)*d/dx P(n-2,x) + (15-4*x)*P(n-1,x) + (4-4*x)*d/dx P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 6 + x, P(3,x) = 22 + 9*x, P(4,x) = 85 + 53*x + x^2, P(5,x) = 322 + 273*x + 20*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 6 + x;
P[3, x_] := 22 + 9*x;
P[4, x_] := 85 + 53*x + x^2;
P[5, x_] := 322 + 273*x + 20*x^2;
P[n_, x_] := P[n, x] = Expand[(7 - 9*x - 3*x^2 + 5*x^3)*P[n - 6, x] - (4 - 12*x + 12*x^2 - 4*x^3)*D[P[n - 6, x], x] + (31 - 22*x + 11*x^2 - 4*x^3)*P[n - 5, x] + (28 - 44*x + 20*x^2 - 4*x^3)*D[P[n - 5, x], x] - (122 - 85*x + 19*x^2)*P[n - 4, x] - (68 - 84*x + 16*x^2)*D[P[n - 4, x], x] + (143 - 78*x + 8*x^2)*P[n - 3, x] + (68 - 76*x + 8*x^2)*D[P[n - 3, x], x] - (74 - 31*x)*P[n - 2, x] - (28 - 28*x)*D[P[n - 2, x], x] + (15 - 4*x)*P[n - 1, x] + (4 - 4*x)*D[P[n - 1, x], 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] = 2; v[3] = 6 + x; v[4] = 22 + 9*x; v[5] = 85 + 53*x + x^2; v[6] = 322 + 273*x + 20*x^2; for(n = 6, N, my(i = n + 1); v[i] = (7 - 9*x - 3*x^2 + 5*x^3)*v[i - 6] - (4 - 12*x + 12*x^2 - 4*x^3)*deriv(v[i - 6], x) + (31 - 22*x + 11*x^2 - 4*x^3)*v[i - 5] + (28 - 44*x + 20*x^2 - 4*x^3)*deriv(v[i - 5], x) - (122 - 85*x + 19*x^2)*v[i - 4] - (68 - 84*x + 16*x^2)*deriv(v[i - 4], x) + (143 - 78*x + 8*x^2)*v[i - 3] + (68 - 76*x + 8*x^2)*deriv(v[i - 3], x) - (74 - 31*x)*v[i - 2] - (28 - 28*x)*deriv(v[i - 2], x) + (15 - 4*x)*v[i - 1] + (4 - 4*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 - 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. [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
  4. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  5. [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
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  7. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023