Real-rooted atlas

A093375 → D → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(4\,n + 8\,n x + 4\,n x^{2} - 4\,n^{2} - 8\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-8\,n - 8\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
26410
324361272
48019214432448
5240800960480802560
6672288048003840144019213824
717929408201602240013440403244871680
8460828672752641075208960043008107521024360448
911520829442580484515844838403225601290242764823041769472
102816023040082944017203202257920193536010752003686406912051208519680
116758461952025344006082560946176099348487096320337920010137601689601126440370176
1215974416220167434240202752003649536045416448397393922433024010137600270336040550424576188743680
13372736415334421086208644300801317888001897758721968046081476034567907328029286400702873695846453248872415232
148601601043660858146816196804608451010560738017280885620736787218432516612096246005760820019201789132822364161146883992977408
Technical details
Node id
n-727a5b76084f597f · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-72ae8e6419635e63

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     1;
  2:     6,     4;
  3:    24,    36,     12;
  4:    80,   192,    144,     32;
  5:   240,   800,    960,    480,     80;
  6:   672,  2880,   4800,   3840,   1440,    192;
  7:  1792,  9408,  20160,  22400,  13440,   4032,    448;
  8:  4608, 28672,  75264, 107520,  89600,  43008,  10752,  1024;
  9: 11520, 82944, 258048, 451584, 483840, 322560, 129024, 27648, 2304;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(4*n^2-4*n+(8*n^2-8*n)*x+(4*n^2-4*n)*x^2)*P(n-2,x) + (4*n^2-8*n+(4*n^2-8*n)*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 6 + 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 4*n + (8*n^2 - 8*n)*x + (4*n^2 - 4*n)*x^2)*P[n - 2, x] + (4*n^2 - 8*n + (4*n^2 - 8*n)*x)*P[n - 1, x])/(n^2 - 3*n + 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; v[3] = 6 + 4*x; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 4*n + (8*n^2 - 8*n)*x + (4*n^2 - 4*n)*x^2)*v[i - 2] + (4*n^2 - 8*n + (4*n^2 - 8*n)*x)*v[i - 1])/(n^2 - 3*n + 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.