Real-rooted atlas

A071951 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 - 4\,x - x^{2}\right)\,P_{n-2}(x) + \left(-12\,x - 5\,x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} - x^{3}\right)\,P_{n-2}''(x) + \left(8 + 2\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
38210
45240395
532058412041028
6193676483276280513145
7116489564876992127685606197622
8699521171072168331248569639480100873450527
941984014192768354228481681529622652801038241680869221544
1025192961711528967297461125516893441161608808532480242088264091580045745
11151162882058873344148516515841752367513655683686406172119362738489651427239601040662800066
1290698752247367127043001213416965455124561922557760768004124591078426917931047758150410141565720111170253591423
1354419456029702194995260395319029761676553547468811424485795840261670229606424192933273610058670336198836352188188080081237396010343404
143265171456356535178854412123617098444851101877344460850078532276864016024471153459220471731302400118177099622433188002848469337440331931610920131318540758661449
151959103692842790751805440243007144737177615492211431317504216717570230886409574646233259520166091681722982412975565400576049801726840329882810080010340844805608512145601451007248989607790
Technical details
Node id
n-a8ea30cabe0c908d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-728f8e328f25a302

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       0;
   2:       1;
   3:       8,         2;
   4:      52,        40,         3;
   5:     320,       584,       120,         4;
   6:    1936,      7648,      3276,       280,         5;
   7:   11648,     95648,     76992,     12768,       560,       6;
   8:   69952,   1171072,   1683312,    485696,     39480,    1008,      7;
   9:  419840,  14192768,  35422848,  16815296,   2265280,  103824,   1680,    8;
  10: 2519296, 171152896, 729746112, 551689344, 116160880, 8532480, 242088, 2640, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(12+4*x+x^2)*P(n-2,x) - (12*x+5*x^2)*d/dx P(n-2,x) - (2*x^2+x^3)*d^2/dx^2 P(n-2,x) + (8+2*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(12 + 4*x + x^2)*P[n - 2, x] - (12*x + 5*x^2)*D[P[n - 2, x], x] - (2*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (8 + 2*x)*P[n - 1, x] + 6*x*D[P[n - 1, x], x] + x^2*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] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(12 + 4*x + x^2)*v[i - 2] - (12*x + 5*x^2)*deriv(v[i - 2], x) - (2*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (8 + 2*x)*v[i - 1] + 6*x*deriv(v[i - 1], x) + x^2*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

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.