Real-rooted atlas

A013620 → Veronese 2 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-128 - 96\,x + 128\,n + 80\,n x + 192\,n^{2} x - 18\,n^{2} x^{2} - 176\,n^{3} x - 18\,n^{3} x^{2} + 18\,n^{4} x^{2} + 18\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-256 - 2496\,x + 144\,x^{2} - 324\,x^{3} + 256\,n + 2464\,n x + 2232\,n x^{2} - 648\,n x^{3} + 384\,n^{2} x - 1044\,n^{2} x^{2} - 81\,n^{2} x^{3} - 352\,n^{3} x - 1404\,n^{3} x^{2} + 567\,n^{3} x^{3} + 36\,n^{4} x^{2} + 405\,n^{4} x^{3} + 36\,n^{5} x^{2} + 81\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(512\,x - 1408\,x^{2} + 864\,x^{3} - 648\,x^{4} - 1344\,n x^{2} + 3888\,n x^{3} - 1944\,n x^{4} - 704\,n^{2} x^{2} + 2520\,n^{2} x^{3} - 2106\,n^{2} x^{4} + 432\,n^{3} x^{3} - 972\,n^{3} x^{4} + 72\,n^{4} x^{3} - 162\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3Pn(1)
011
122
243640
38216224
4168647291609
5322880729010202
6648640437401166464108
712824192204120163296391736

All 15 rows

Technical details
Node id
n-7786a9e5eec1e3a1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1e38082c567e7267

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:   4,     36;
  3:   8,    216;
  4:  16,    864,     729;
  5:  32,   2880,    7290;
  6:  64,   8640,   43740,   11664;
  7: 128,  24192,  204120,  163296;
  8: 256,  64512,  816480, 1306368,  164025;
  9: 512, 165888, 2939328, 7838208, 2952450;
%F Formula
The row polynomials satisfy (128*n-128-(176*n^3-192*n^2-80*n+96)*x+(18*n^5+18*n^4-18*n^3-18*n^2)*x^2)*P(n,x) = (256*n-256-(352*n^3-384*n^2-2464*n+2496)*x+(36*n^5+36*n^4-1404*n^3-1044*n^2+2232*n+144)*x^2+(81*n^5+405*n^4+567*n^3-81*n^2-648*n-324)*x^3)*P(n-1,x) + (512*x-(704*n^2+1344*n+1408)*x^2+(72*n^4+432*n^3+2520*n^2+3888*n+864)*x^3-(162*n^4+972*n^3+2106*n^2+1944*n+648)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((256*n - 256 - (352*n^3 - 384*n^2 - 2464*n + 2496)*x + (36*n^5 + 36*n^4 - 1404*n^3 - 1044*n^2 + 2232*n + 144)*x^2 + (81*n^5 + 405*n^4 + 567*n^3 - 81*n^2 - 648*n - 324)*x^3)*P[n - 1, x] + (512*x - (704*n^2 + 1344*n + 1408)*x^2 + (72*n^4 + 432*n^3 + 2520*n^2 + 3888*n + 864)*x^3 - (162*n^4 + 972*n^3 + 2106*n^2 + 1944*n + 648)*x^4)*D[P[n - 1, x], x])/(128*n - 128 - (176*n^3 - 192*n^2 - 80*n + 96)*x + (18*n^5 + 18*n^4 - 18*n^3 - 18*n^2)*x^2)]];
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; for(n = 2, N, my(i = n + 1); v[i] = ((256*n - 256 - (352*n^3 - 384*n^2 - 2464*n + 2496)*x + (36*n^5 + 36*n^4 - 1404*n^3 - 1044*n^2 + 2232*n + 144)*x^2 + (81*n^5 + 405*n^4 + 567*n^3 - 81*n^2 - 648*n - 324)*x^3)*v[i - 1] + (512*x - (704*n^2 + 1344*n + 1408)*x^2 + (72*n^4 + 432*n^3 + 2520*n^2 + 3888*n + 864)*x^3 - (162*n^4 + 972*n^3 + 2106*n^2 + 1944*n + 648)*x^4)*deriv(v[i - 1], x))/(128*n - 128 - (176*n^3 - 192*n^2 - 80*n + 96)*x + (18*n^5 + 18*n^4 - 18*n^3 - 18*n^2)*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. [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.