Real-rooted atlas

A038220 → Veronese 2 (j=1) → D+1

nonnegative coefficients

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

Recurrence

\(\left(-117 + 138\,x - 24\,x^{2} + 147\,n - 162\,n x + 24\,n x^{2} - 36\,n^{2} + 24\,n^{2} x + 6\,n^{3}\right)\,P_{n}(x) = \left(-136\,x + 304\,x^{2} - 64\,x^{3} + 279\,n - 266\,n x - 144\,n x^{2} + 32\,n x^{3} - 54\,n^{2} + 16\,n^{2} x + 32\,n^{2} x^{2} + 18\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(234\,x - 644\,x^{2} + 384\,x^{3} - 64\,x^{4} - 108\,n x + 192\,n x^{2} - 64\,n x^{3} + 36\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
21212
362870
431296408
51530784322346
67236547257613284
73288634776643212874222
8143856205632576003072410160
96079861143072451008435205122246098
102493180601862432037124761601536012207036
11995959830215592210418564435200263680204865917974
123889360814549673612933043236799488340992073728354003912
1314892157867552056075057840027873331236840960148684881921892089850
145604931083037795488414319276819570636803497195522219212834406410070800788
Technical details
Node id
n-79b9f4e804ca7a21 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5b1c89fbf9485e53

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:      2;
  2:     12;
  3:     62,       8;
  4:    312,      96;
  5:   1530,     784,     32;
  6:   7236,    5472,    576;
  7:  32886,   34776,   6432,   128;
  8: 143856,  205632,  57600,  3072;
  9: 607986, 1143072, 451008, 43520, 512;
%F Formula
The row polynomials satisfy (6*n^3-36*n^2+147*n-117+(24*n^2-162*n+138)*x+(24*n-24)*x^2)*P(n,x) = (18*n^3-54*n^2+279*n+(8*n^3+16*n^2-266*n-136)*x+(32*n^2-144*n+304)*x^2+(32*n-64)*x^3)*P(n-1,x) + ((36*n^2-108*n+234)*x-(16*n^2-192*n+644)*x^2-(64*n-384)*x^3-64*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^3 - 54*n^2 + 279*n + (8*n^3 + 16*n^2 - 266*n - 136)*x + (32*n^2 - 144*n + 304)*x^2 + (32*n - 64)*x^3)*P[n - 1, x] + ((36*n^2 - 108*n + 234)*x - (16*n^2 - 192*n + 644)*x^2 - (64*n - 384)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(6*n^3 - 36*n^2 + 147*n - 117 + (24*n^2 - 162*n + 138)*x + (24*n - 24)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((18*n^3 - 54*n^2 + 279*n + (8*n^3 + 16*n^2 - 266*n - 136)*x + (32*n^2 - 144*n + 304)*x^2 + (32*n - 64)*x^3)*v[i - 1] + ((36*n^2 - 108*n + 234)*x - (16*n^2 - 192*n + 644)*x^2 - (64*n - 384)*x^3 - 64*x^4)*deriv(v[i - 1], x))/(6*n^3 - 36*n^2 + 147*n - 117 + (24*n^2 - 162*n + 138)*x + (24*n - 24)*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.