Real-rooted atlas

A007318 → D+1 → D+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
276114
313219144
4215242121128
53110513070151352
643186315260105181928
7573016517354551472112368
8734561204173614707281962415888
9916572052361239062646109225227114336
1011191032856840903078124410156031530134304
1113312215005120451881019866143226930214538533180896
12157159673262002036135451443973224552103952860462361188416
13183204110374317466506593951978127378839897150153718546391434176
142112562142874841211111118218221921919562412912962062210214732637421991232

Generated b-files: A007466 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-3d02bb5acf6eb220 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d19c49978a21a529

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  3,   1;
  2:  7,   6,    1;
  3: 13,  21,    9,    1;
  4: 21,  52,   42,   12,    1;
  5: 31, 105,  130,   70,   15,    1;
  6: 43, 186,  315,  260,  105,   18,    1;
  7: 57, 301,  651,  735,  455,  147,   21,   1;
  8: 73, 456, 1204, 1736, 1470,  728,  196,  24,  1;
  9: 91, 657, 2052, 3612, 3906, 2646, 1092, 252, 27, 1;
%F Formula
The row polynomials satisfy (n^2-n+1+2*n*x+x^2)*P(n,x) = (n^2+n+1+(n^2+3*n+3)*x+(2*n+3)*x^2+x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + 1 + (n^2 + 3*n + 3)*x + (2*n + 3)*x^2 + x^3)*P[n - 1, x])/(n^2 - n + 1 + 2*n*x + 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; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + n + 1 + (n^2 + 3*n + 3)*x + (2*n + 3)*x^2 + x^3)*v[i - 1])/(n^2 - n + 1 + 2*n*x + 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-31ecd8005cc9acb4
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-31ecd8005cc9acb4
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-31ecd8005cc9acb4

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [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
  5. [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
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.