Real-rooted atlas

A038207 → Veronese 2 (j=1) → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-128 + 32\,x + 128\,n - 176\,n x + 224\,n^{2} x - 2\,n^{2} x^{2} - 80\,n^{3} x + 6\,n^{3} x^{2} - 6\,n^{4} x^{2} + 2\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-512\,x - 64\,x^{2} + 256\,n + 192\,n x + 48\,n x^{2} + 288\,n^{2} x + 136\,n^{2} x^{2} - 2\,n^{2} x^{3} - 160\,n^{3} x - 68\,n^{3} x^{2} - 3\,n^{3} x^{3} - 8\,n^{4} x^{2} + 4\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(512\,x - 32\,x^{3} - 64\,n x^{2} + 16\,n x^{3} - 320\,n^{2} x^{2} + 88\,n^{2} x^{3} - 2\,n^{2} x^{4} + 16\,n^{3} x^{3} - 4\,n^{3} x^{4} + 8\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
244
312416
4323264
5801609249
6192640108940
74482240756163460
810247168403225612480
92304215041814423042544281
105120614407257615360500154996
111126416896026611284480550036536352
1224576450560912384405504440008641837888
135324811714562965248175718428600011232496244417
141146882981888922521670287361601600104832137221058332
Technical details
Node id
n-7b5f371f6c6f4176 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-74ce1bee19a36935

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (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:    1;
  2:    4;
  3:   12,     4;
  4:   32,    32;
  5:   80,   160,     9;
  6:  192,   640,   108;
  7:  448,  2240,   756,   16;
  8: 1024,  7168,  4032,  256;
  9: 2304, 21504, 18144, 2304, 25;
%F Formula
The row polynomials satisfy (128*n-128-(80*n^3-224*n^2+176*n-32)*x+(2*n^5-6*n^4+6*n^3-2*n^2)*x^2)*P(n,x) = (256*n-(160*n^3-288*n^2-192*n+512)*x+(4*n^5-8*n^4-68*n^3+136*n^2+48*n-64)*x^2+(n^5-3*n^3-2*n^2)*x^3)*P(n-1,x) + (512*x-(320*n^2+64*n)*x^2+(8*n^4+16*n^3+88*n^2+16*n-32)*x^3-(2*n^4+4*n^3+2*n^2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((256*n - (160*n^3 - 288*n^2 - 192*n + 512)*x + (4*n^5 - 8*n^4 - 68*n^3 + 136*n^2 + 48*n - 64)*x^2 + (n^5 - 3*n^3 - 2*n^2)*x^3)*P[n - 1, x] + (512*x - (320*n^2 + 64*n)*x^2 + (8*n^4 + 16*n^3 + 88*n^2 + 16*n - 32)*x^3 - (2*n^4 + 4*n^3 + 2*n^2)*x^4)*D[P[n - 1, x], x])/(128*n - 128 - (80*n^3 - 224*n^2 + 176*n - 32)*x + (2*n^5 - 6*n^4 + 6*n^3 - 2*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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((256*n - (160*n^3 - 288*n^2 - 192*n + 512)*x + (4*n^5 - 8*n^4 - 68*n^3 + 136*n^2 + 48*n - 64)*x^2 + (n^5 - 3*n^3 - 2*n^2)*x^3)*v[i - 1] + (512*x - (320*n^2 + 64*n)*x^2 + (8*n^4 + 16*n^3 + 88*n^2 + 16*n - 32)*x^3 - (2*n^4 + 4*n^3 + 2*n^2)*x^4)*deriv(v[i - 1], x))/(128*n - 128 - (80*n^3 - 224*n^2 + 176*n - 32)*x + (2*n^5 - 6*n^4 + 6*n^3 - 2*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

Same fingerprint ?

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.