Real-rooted atlas

A397048 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 2\,x + 2\,x^{2} + 3\,n x - 4\,n x^{2} + 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x^{2} + 4\,x^{3} - 3\,n x^{3}\right)\,P_{n-1}'(x) + \left(x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
215410
3112372777
4122151369256799
513542521904651312510427
61519658550367517099346656163967
717019042583018650170610412736098235433017562
8192340265562714021445363815341887262695051677721663625324
9111756461466642245467208535251164194653726840906125795113874204891512354975
1011458850298080610442778962625648965605331977883010021257511159374246011000000000040012800675
11117613255561825148308272548359662893205722215723222001027995715012957276616364576967000772853116706111166271373797
1212101912999643532947827725834868108779531081112161166167657024732613440459280654950692128929514381984674225891610044825637134022033885
131247267671680822680456921869648066357471468644797009817604498492091093289735811314951238806059104253308203142216704908396666618913028751065922531282405154139046
14128736491271853413226012844336992701053096358601794721969040219753181778131916233032383671166163679001685477863594937183012391874406352351180809191998326091111200682555801647745103281852282
Technical details
Node id
n-441f182f6bd20284 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f13747014c7999ad

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   5,    4;
  3: 1,  12,   37,     27;
  4: 1,  22,  151,    369,     256;
  5: 1,  35,  425,   2190,    4651,     3125;
  6: 1,  51,  965,   8550,   36751,    70993,     46656;
  7: 1,  70, 1904,  25830,  186501,   706104,   1273609,    823543;
  8: 1,  92, 3402,  65562,  714021,  4453638,  15341887,  26269505,  16777216;
  9: 1, 117, 5646, 146664, 2245467, 20853525, 116419465, 372684090, 612579511, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (1+(3*n-2)*x+(2*n^2-4*n+2)*x^2)*P(n-1,x) - (2*x^2+(3*n-4)*x^3)*d/dx P(n-1,x) + x^4*d^2/dx^2 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[(1 + (3*n - 2)*x + (2*n^2 - 4*n + 2)*x^2)*P[n - 1, x] - (2*x^2 + (3*n - 4)*x^3)*D[P[n - 1, x], x] + x^4*D[P[n - 1, x], {x, 2}]];
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] = (1 + (3*n - 2)*x + (2*n^2 - 4*n + 2)*x^2)*v[i - 1] - (2*x^2 + (3*n - 4)*x^3)*deriv(v[i - 1], x) + x^4*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 - 1, v[i])))};
T(9)

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.