Real-rooted atlas

A137320: Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.

A137320 nonnegative coefficients only rational roots

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

Recurrence from OEIS A137320
\(P_{n}(x) = \left(-1 + 2\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1022
20246
30412824
4012444816120
504820028016032720
60240109618001360480645040
701440705612992117605600134412840320

Show all 15 stored rows

Generated b-files: A125553, A137320 ?

Computed in your browser ?

Technical details
Node id
n-60c7e36cbe887b7e · revision 1 · JSON
Definition source
oeis_catalog:A137320
Fingerprint
f6-21d4cbc8ebff768c

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): yes

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,     2;
  2: 0,     2,      4;
  3: 0,     4,     12,      8;
  4: 0,    12,     44,     48,      16;
  5: 0,    48,    200,    280,     160,     32;
  6: 0,   240,   1096,   1800,    1360,    480,     64;
  7: 0,  1440,   7056,  12992,   11760,   5600,   1344,   128;
  8: 0, 10080,  52272, 105056,  108304,  62720,  20608,  3584,  256;
  9: 0, 80640, 438336, 944992, 1076544, 718368, 290304, 69888, 9216, 512;
%F Formula
The row polynomials satisfy P(n,x) = (n-1+2*x)*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[(n - 1 + 2*x)*P[n - 1, 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] = (n - 1 + 2*x)*v[i - 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)

Generated with polytool c48629374597.

A125553: triangle, offset 1, from the row P1, low zeros dropped
%e Example
Triangle begins:
   1:      2;
   2:      2,       4;
   3:      4,      12,       8;
   4:     12,      44,      48,       16;
   5:     48,     200,     280,      160,      32;
   6:    240,    1096,    1800,     1360,     480,      64;
   7:   1440,    7056,   12992,    11760,    5600,    1344,     128;
   8:  10080,   52272,  105056,   108304,   62720,   20608,    3584,    256;
   9:  80640,  438336,  944992,  1076544,  718368,  290304,   69888,   9216,   512;
  10: 725760, 4106304, 9381600, 11578880, 8618400, 4049472, 1209600, 222720, 23040, 1024;
%F Formula
Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (n-1+2*x)*P(n-1,x) for n >= 2, with P(1,x) = 2*x.
%t Mathematica
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[(n - 1 + 2*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2*x; for(n = 2, N, my(i = n); v[i] = (n - 1 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A137320: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 0,     2;
  2: 0,     2,      4;
  3: 0,     4,     12,      8;
  4: 0,    12,     44,     48,      16;
  5: 0,    48,    200,    280,     160,     32;
  6: 0,   240,   1096,   1800,    1360,    480,     64;
  7: 0,  1440,   7056,  12992,   11760,   5600,   1344,   128;
  8: 0, 10080,  52272, 105056,  108304,  62720,  20608,  3584,  256;
  9: 0, 80640, 438336, 944992, 1076544, 718368, 290304, 69888, 9216, 512;
%F Formula
The row polynomials satisfy P(n,x) = (n-1+2*x)*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[(n - 1 + 2*x)*P[n - 1, 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] = (n - 1 + 2*x)*v[i - 1]); 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, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-6aa1e3cc7e408461
  • 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-6aa1e3cc7e408461
  • 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-6aa1e3cc7e408461
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: (-inf, 0] -> (-inf, 0] through R_s, s = 1, and U_g
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-e739032c2ab562e9

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 14 proof steps

Same fingerprint (3)

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

References

  1. [Bre89] Francesco Brenti. Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc., 81(413):viii+106, 1989. doi:10.1090/memo/0413
  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