Real-rooted atlas

A216973 → θ+1 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-2 + 2\,x - n x + 2\,n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
26410
32430963
4801448416324
5240560480180251485
6672192021601200330366318
7179260488400616025205464925515
846081792029568268801456047048406499144
911520506889676810483270560302408064122481373977
1028160138240299520376320302400161280571201296017101001377810
1167584366080887040126720011827207539843326401003201980023101214979799
121597449461762534400405504043084803193344168537663360016632029040303614417714700
133727362396160702873612446720148262401251993676876803459456113256026312041184390016962178597
14860160596377619009536369008644868864046126080322882561691289666306241921920400400567844914196215765046

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(4\,n + 4\,n x + n x^{2} - 4\,n^{2} - 4\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-8\,n - 4\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 4\,x\)
Technical details
Node id
n-e67be8a32bf8f89b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-f2bcce74662ca3cf)
Fingerprint
f6-48023a6581f41e60

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,    30,     9;
  4:    80,   144,    84,     16;
  5:   240,   560,   480,    180,    25;
  6:   672,  1920,  2160,   1200,   330,    36;
  7:  1792,  6048,  8400,   6160,  2520,   546,   49;
  8:  4608, 17920, 29568,  26880, 14560,  4704,  840,   64;
  9: 11520, 50688, 96768, 104832, 70560, 30240, 8064, 1224, 81;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (2*n^2-2+(n^2-n+2)*x)*P(n-1,x) + (2*x+x^2)*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[((2*n^2 - 2 + (n^2 - n + 2)*x)*P[n - 1, x] + (2*x + x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
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] = ((2*n^2 - 2 + (n^2 - n + 2)*x)*v[i - 1] + (2*x + x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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.