Real-rooted atlas

Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.

A038337 (tail) nonnegative coefficients only rational roots

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

Recurrence from OEIS A038337
\(P_{n}(x) = \left(12 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 264\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0144264121529
1172847524356133112167
220736760321045446388814641279841
32488321140480209088019166408784601610516436343
4298598416422912376358404599936031624560115956721771561148035889
535831808229920768632282112965986560885487680487018224148811124194871713404825447
64299816963153199104101165137921854694195221251704320155845831687142933952187076841621435888178310985281
75159780352425681879041560833556483338449551364590368133124207837455362571456222721010214944642315075914823579476911801152661463

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-344e4bfa9933088a · revision 2 · JSON
Definition source
oeis_catalog:A038337
Fingerprint
f6-60cc3cc9b4461a16

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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 ?

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:          144,           264,            121;
  1:         1728,          4752,           4356,           1331;
  2:        20736,         76032,         104544,          63888,           14641;
  3:       248832,       1140480,        2090880,        1916640,          878460,          161051;
  4:      2985984,      16422912,       37635840,       45999360,        31624560,        11595672,         1771561;
  5:     35831808,     229920768,      632282112,      965986560,       885487680,       487018224,       148811124,        19487171;
  6:    429981696,    3153199104,    10116513792,    18546941952,     21251704320,     15584583168,      7142933952,      1870768416,      214358881;
  7:   5159780352,   42568187904,   156083355648,   333844955136,    459036813312,    420783745536,    257145622272,    101021494464,    23150759148,     2357947691;
  8:  61917364224,  567575838720,  2341250334720,  5723056373760,   9180736266240,  10098809892864,   7714368668160,   4040859778560,  1389045548880,   282953722920,   25937424601;
  9: 743008370688, 7492001071104, 34338338242560, 94430430167040, 173122455306240, 222173817643008, 203659332839424, 133348372692480, 61118004150720, 18674945712720, 3423740047332, 285311670611;
%F Formula
The row polynomials satisfy P(n,x) = (12+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 264*x + 121*x^2.
%t Mathematica
P[0, x_] := 144 + 264*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 11*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] = 144 + 264*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 11*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.

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-37ab130d55b994ea
  • 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-107a218257e6f127, e-37ab130d55b994ea, e-9aedabfb67a4638b
  • 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-107a218257e6f127, e-37ab130d55b994ea, e-9aedabfb67a4638b
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-37ab130d55b994ea
  • 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-107a218257e6f127, e-9aedabfb67a4638b
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times 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-832e27a12accd5ee
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 12] to roots in (-inf, 0]: an input root r <= 12 gives the root (r - a)/d <= 0; a larger root gives a positive root
    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-832e27a12accd5ee

Proof chains

4 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 7 proof steps

Same fingerprint (3)

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