Real-rooted atlas

A390433 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(1 + 16\,x - 16\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(9\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(x^{3}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
111
211
3145
41203657
51565365761169
611203280198721440037673
71220130002297929881285184001749541
81364396201529792196012806375859225401600110331249
9156010113672350722047061122023886592518984755216257024009051479425
10181622713627057728142743328031501566720250080125184520610457600131681894400935328762865
11111404629608520454475154220803112784896005573844900864365321484840966312554726400013168189440000118718609669285
121154087450023520358432139535296223522654208075611505060864112900574905344062370586410762249103246995456000159335092224000018140541415043529
131202415536405850035841170480291201264843597798472159197572198420485551044634624260263352178100224123156674064639590415400420931543040002294425328025600003282534627872283089
14126002624336133696534437503460966459467647625984528767736377420825428135117855744061805876040775065606785898489824005324827864255873859338240030207769279104614400038775788043632640000693795240948153885785
15132764249336284858288010822892766402409843987033603151290996683315223693522966882406409797615537953674240020703693686193521786881988988919432996877107271634572729870952038400680062566660290150400007600054456551997440000169301519678057018099845
Technical details
Node id
n-d77a552bbe571714 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-17ce5c7cd2025ccd

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1;
   3: 1,   4;
   4: 1,  20,     36;
   5: 1,  56,    536,      576;
   6: 1, 120,   3280,    19872,      14400;
   7: 1, 220,  13000,   229792,     988128,      518400;
   8: 1, 364,  39620,  1529792,   19601280,    63758592,     25401600;
   9: 1, 560, 101136,  7235072,  204706112,  2023886592,   5189847552,   1625702400;
  10: 1, 816, 227136, 27057728, 1427433280, 31501566720, 250080125184, 520610457600, 131681894400;
%F Formula
The row polynomials satisfy P(n,x) = (1+(4*n^2-16*n+16)*x)*P(n-1,x) - (4*n-9)*x^2*d/dx P(n-1,x) + x^3*d^2/dx^2 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[(1 + (4*n^2 - 16*n + 16)*x)*P[n - 1, x] - (4*n - 9)*x^2*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}]];
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] = (1 + (4*n^2 - 16*n + 16)*x)*v[i - 1] - (4*n - 9)*x^2*deriv(v[i - 1], x) + x^3*deriv(deriv(v[i - 1], x), 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)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.