Real-rooted atlas

Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.

A038336 (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 A038336
\(P_{n}(x) = \left(12 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 240\,x + 100\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0144240100484
1172843203600100010648
22073669120864004800010000234256
32488321036800172800014400006000001000005153632
429859841492992031104000345600002160000072000001000000113379904
53583180820901888052254720072576000060480000030240000084000000100000002494357888
64299816962866544640836075520013934592000145152000009676800000403200000096000000010000000054875873536
7515978035238698352640128994508800250822656000313528320000261273600000145152000000518400000001080000000010000000001207269217792

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-ec89fb3fc20684fc · revision 1 · JSON
Definition source
oeis_catalog:A038336
Fingerprint
f6-da2debf48cb6165b

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,           240,            100;
  1:         1728,          4320,           3600,           1000;
  2:        20736,         69120,          86400,          48000,           10000;
  3:       248832,       1036800,        1728000,        1440000,          600000,          100000;
  4:      2985984,      14929920,       31104000,       34560000,        21600000,         7200000,         1000000;
  5:     35831808,     209018880,      522547200,      725760000,       604800000,       302400000,        84000000,       10000000;
  6:    429981696,    2866544640,     8360755200,    13934592000,     14515200000,      9676800000,      4032000000,      960000000,      100000000;
  7:   5159780352,   38698352640,   128994508800,   250822656000,    313528320000,    261273600000,    145152000000,    51840000000,    10800000000,    1000000000;
  8:  61917364224,  515978035200,  1934917632000,  4299816960000,   6270566400000,   6270566400000,   4354560000000,  2073600000000,   648000000000,  120000000000,   10000000000;
  9: 743008370688, 6810910064640, 28378791936000, 70946979840000, 118244966400000, 137952460800000, 114960384000000, 68428800000000, 28512000000000, 7920000000000, 1320000000000, 100000000000;
%F Formula
The row polynomials satisfy P(n,x) = (12+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 240*x + 100*x^2.
%t Mathematica
P[0, x_] := 144 + 240*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 10*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 + 240*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 10*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-2f1e093ae220e593
  • 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-2f1e093ae220e593, e-3f0a5f7ceded39cb
  • 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-2f1e093ae220e593, e-3f0a5f7ceded39cb
  • 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-2f1e093ae220e593
  • 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-3f0a5f7ceded39cb
  • 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-8872d945c5d0b661
  • 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-8872d945c5d0b661

Proof chains

3 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 (6)

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