Real-rooted atlas

A341287: Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise).

A023531 → basis A341287

A341287

Proved for all n: all roots real, in (−∞, 1] · interlacing (weak) what this means

Recurrence from OEIS A341287
\(P_{n}(x) = \left(x\right)\,P_{n-1}(x) + \left(-1 + x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1011
2-1111
3-1-2311
42-91611
59-9-25151011
6950-104-20501511
7-50267-98-3641051192111
8-2674131163-1610-5395742382811
9-413-21807569-1511-663690318064263611
102180-177311749129580-32570-12957875745007054511
1117731-50533-71489236511-35915-149897808534617973511005511
1250533110176-903044781704842336-833910-3437061725241025971908516396611
13-1101761966797-4041024-19272768320598-1264978-4103814431868215352579723474923537811
14-19667979938669-333423-431052443768000626618592-26190164-10373792429371827957935795795969632769111

Generated b-files: A341287 ?

Computed in your browser ?

Technical details
Node id
n-13114c927e22057d · revision 1 · JSON
Definition source
oeis_catalog:A341287
Fingerprint
f6-97fb096e972a926e

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): no
  • 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:    0,     1;
  2:   -1,     1,    1;
  3:   -1,    -2,    3,     1;
  4:    2,    -9,    1,     6,     1;
  5:    9,    -9,  -25,    15,    10,   1;
  6:    9,    50, -104,   -20,    50,  15,    1;
  7:  -50,   267,  -98,  -364,   105, 119,   21,   1;
  8: -267,   413, 1163, -1610,  -539, 574,  238,  28,  1;
  9: -413, -2180, 7569, -1511, -6636, 903, 1806, 426, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) - (1-x)*d/dx 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[x*P[n - 1, x] - (1 - x)*D[P[n - 1, x], x]];
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] = x*v[i - 1] - (1 - x)*deriv(v[i - 1], x)); 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.

A341287: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:    1;
  1:    0,     1;
  2:   -1,     1,    1;
  3:   -1,    -2,    3,     1;
  4:    2,    -9,    1,     6,     1;
  5:    9,    -9,  -25,    15,    10,   1;
  6:    9,    50, -104,   -20,    50,  15,    1;
  7:  -50,   267,  -98,  -364,   105, 119,   21,   1;
  8: -267,   413, 1163, -1610,  -539, 574,  238,  28,  1;
  9: -413, -2180, 7569, -1511, -6636, 903, 1806, 426, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) - (1-x)*d/dx 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[x*P[n - 1, x] - (1 - x)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = x*v[i - 1] - (1 - x)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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, 1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-0f172f031b36afc5, e-73143461db750abd
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-73143461db750abd

Proof chains

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

Used in 2 proof steps

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. [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
  4. [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
  5. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741