Real-rooted atlas

A159854 → Veronese 2 (j=1) → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-144 + 4\,x - 4\,x^{2} + 144\,n + 62\,n x - 8\,n x^{2} - 28\,n^{2} x - n^{2} x^{2} - 38\,n^{3} x + 7\,n^{3} x^{2} + 5\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-288 - 144\,x + 144\,n + 604\,n x - 148\,n x^{2} + 36\,n x^{3} + 10\,n^{2} x - 228\,n^{2} x^{2} + 60\,n^{2} x^{3} - 38\,n^{3} x - 61\,n^{3} x^{2} + 37\,n^{3} x^{3} + 4\,n^{4} x^{2} + 10\,n^{4} x^{3} + n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(288\,x - 504\,x^{2} + 288\,x^{3} - 72\,x^{4} - 284\,n x^{2} + 404\,n x^{3} - 120\,n x^{4} - 76\,n^{2} x^{2} + 150\,n^{2} x^{3} - 74\,n^{2} x^{4} + 20\,n^{3} x^{3} - 20\,n^{3} x^{4} + 2\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\,x\)

Rows ?

nx0x1x2x3x4Pn(1)
00
1044
2088
3012921
40163652
50209016126
602418096300
702831533625704

All 15 rows

Technical details
Node id
n-392d05dc51c23e82 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-30123651382d8f56

Observed on the stored rows ?

  • Degrees: -1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 0;
  1: 0,  4;
  2: 0,  8;
  3: 0, 12,   9;
  4: 0, 16,  36;
  5: 0, 20,  90,   16;
  6: 0, 24, 180,   96;
  7: 0, 28, 315,  336,  25;
  8: 0, 32, 504,  896, 200;
  9: 0, 36, 756, 2016, 900, 36;
%F Formula
The row polynomials satisfy (144*n-144-(38*n^3+28*n^2-62*n-4)*x+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^2)*P(n,x) = (144*n-288-(38*n^3-10*n^2-604*n+144)*x+(n^5+4*n^4-61*n^3-228*n^2-148*n)*x^2+(n^5+10*n^4+37*n^3+60*n^2+36*n)*x^3)*P(n-1,x) + (288*x-(76*n^2+284*n+504)*x^2+(2*n^4+20*n^3+150*n^2+404*n+288)*x^3-(2*n^4+20*n^3+74*n^2+120*n+72)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((144*n - 288 - (38*n^3 - 10*n^2 - 604*n + 144)*x + (n^5 + 4*n^4 - 61*n^3 - 228*n^2 - 148*n)*x^2 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^3)*P[n - 1, x] + (288*x - (76*n^2 + 284*n + 504)*x^2 + (2*n^4 + 20*n^3 + 150*n^2 + 404*n + 288)*x^3 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^4)*D[P[n - 1, x], x])/(144*n - 144 - (38*n^3 + 28*n^2 - 62*n - 4)*x + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^2)]];
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] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((144*n - 288 - (38*n^3 - 10*n^2 - 604*n + 144)*x + (n^5 + 4*n^4 - 61*n^3 - 228*n^2 - 148*n)*x^2 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^3)*v[i - 1] + (288*x - (76*n^2 + 284*n + 504)*x^2 + (2*n^4 + 20*n^3 + 150*n^2 + 404*n + 288)*x^3 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^4)*deriv(v[i - 1], x))/(144*n - 144 - (38*n^3 + 28*n^2 - 62*n - 4)*x + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^2)); 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.