Real-rooted atlas

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

A038338 (tail) palindromic γ-positive 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 A038338
\(P_{n}(x) = \left(12 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 288\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0144288144576
1172851845184172813824
220736829441244168294420736331776
324883212441602488320248832012441602488327962624
4298598417915904447897605971968044789760179159042985984191102976
53583180825082265675246796812541132801254113280752467968250822656358318084586471424
6429981696343985356812039487488240789749763009871872024078974976120394874883439853568429981696110075314176
75159780352464380231681857520926724334215495686501323243526501323243524334215495681857520926724643802316851597803522641807540224

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-c1532953a6736702 · revision 1 · JSON
Definition source
oeis_catalog:A038338
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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,           288,            144;
  1:         1728,          5184,           5184,            1728;
  2:        20736,         82944,         124416,           82944,           20736;
  3:       248832,       1244160,        2488320,         2488320,         1244160,          248832;
  4:      2985984,      17915904,       44789760,        59719680,        44789760,        17915904,         2985984;
  5:     35831808,     250822656,      752467968,      1254113280,      1254113280,       752467968,       250822656,        35831808;
  6:    429981696,    3439853568,    12039487488,     24078974976,     30098718720,     24078974976,     12039487488,      3439853568,       429981696;
  7:   5159780352,   46438023168,   185752092672,    433421549568,    650132324352,    650132324352,    433421549568,    185752092672,     46438023168,     5159780352;
  8:  61917364224,  619173642240,  2786281390080,   7430083706880,  13002646487040,  15603175784448,  13002646487040,   7430083706880,   2786281390080,   619173642240,   61917364224;
  9: 743008370688, 8173092077568, 40865460387840, 122596381163520, 245192762327040, 343269867257856, 343269867257856, 245192762327040, 122596381163520, 40865460387840, 8173092077568, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (12+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 288*x + 144*x^2.
%t Mathematica
P[0, x_] := 144 + 288*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 12*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 + 288*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 12*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-e4b4d500ff341474
  • 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-969ef99525bc6c6a, e-e4b4d500ff341474
  • 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-969ef99525bc6c6a, e-e4b4d500ff341474
  • 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-e4b4d500ff341474
  • 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-969ef99525bc6c6a
  • 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-d2f9900eddbe015a
  • 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-d2f9900eddbe015a

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 8 proof steps

Same fingerprint (59)

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