Real-rooted atlas

A143495 → D+1

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(4 + x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2415
3169126
46461151141
5256369151221799
61024210112753053014736
74096115299751341054539129371
8163846174170035336217770896491190497
965536325089481951305382957811583413866011291020
1026214416909813216795261962510716302382872969420467219131275
1110485768717049209917512155417011192665321621353522752200291085167310847
124194304446338211346675551718709411110954904013858285684831109427871204015991516369838
131677721622736340985263915113377641421060634861472342728125823412207721802154867139315540111414116416797
146710886411535942615343198315102169881459822843030531171941317305768483543173924663058439659622149297111130134051164985
1526843545658350801693321278415176862115330887997323855762831774722617487893562806816091388440098188090697496832159491911471291871399682

Generated b-files: A045379 (row sums), A143496 ?

Computed in your browser ?

Technical details
Node id
n-1f64b8c87ab9b182 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d99d39db05a8c355

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 -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

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      4,       1;
   3:     16,       9,       1;
   4:     64,      61,      15,       1;
   5:    256,     369,     151,      22,       1;
   6:   1024,    2101,    1275,     305,      30,      1;
   7:   4096,   11529,    9751,    3410,     545,     39,     1;
   8:  16384,   61741,   70035,   33621,    7770,    896,    49,    1;
   9:  65536,  325089,  481951,  305382,   95781,  15834,  1386,   60,  1;
  10: 262144, 1690981, 3216795, 2619625, 1071630, 238287, 29694, 2046, 72, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (4 + x)*v[i - 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, v[i])))};
T(10)

Generated with polytool c48629374597.

A143496: triangle, offset 4, from the row P1
%e Example
Triangle begins:
   4:      1;
   5:      4,       1;
   6:     16,       9,       1;
   7:     64,      61,      15,       1;
   8:    256,     369,     151,      22,       1;
   9:   1024,    2101,    1275,     305,      30,      1;
  10:   4096,   11529,    9751,    3410,     545,     39,     1;
  11:  16384,   61741,   70035,   33621,    7770,    896,    49,    1;
  12:  65536,  325089,  481951,  305382,   95781,  15834,  1386,   60,  1;
  13: 262144, 1690981, 3216795, 2619625, 1071630, 238287, 29694, 2046, 72, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 5, with P(4,x) = 1.
%t Mathematica
P[4, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n - 3], {n, 4, 13}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 3)); v[1] = 1; for(n = 5, N, my(i = n - 3); v[i] = (4 + x)*v[i - 1] + x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 3 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 3, v[i])))};
T(13)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-bbdbb484d8de66ad
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-bbdbb484d8de66ad
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-bbdbb484d8de66ad
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-bbdbb484d8de66ad

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

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

References

  1. [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
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.