Real-rooted atlas

A272867 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,x + x^{2} + 6\,n + 2\,n x\right)\,P_{n}(x) = \left(36 + 72\,x + 48\,x^{2} + 12\,x^{3} + x^{4} + 36\,n + 48\,n x + 18\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1128121
210816884161377
3864216020529362162416253
4648022464319682419210584273640832198865
5466562073603952804233602808001200963348060006604011513733
6326592177292842456965909760530712032296321362528400896815401116097248122648873
722394881437004841477184711970568099481664447488369139681543622447295361055376168588186481344561333049821

All 15 rows

Technical details
Node id
n-e706d901432a9ac4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8a253671332bc96a

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:        12,         8,          1;
  2:       108,       168,         84,         16,           1;
  3:       864,      2160,       2052,        936,         216,          24,           1;
  4:      6480,     22464,      31968,      24192,       10584,        2736,         408,         32,          1;
  5:     46656,    207360,     395280,     423360,      280800,      120096,       33480,       6000,        660,         40,         1;
  6:    326592,   1772928,    4245696,    5909760,     5307120,     3229632,     1362528,     400896,      81540,      11160,       972,        48,        1;
  7:   2239488,  14370048,   41477184,   71197056,    80994816,    64447488,    36913968,   15436224,    4729536,    1055376,    168588,     18648,     1344,      56,      1;
  8:  15116544, 111974400,  378473472,  773369856,  1067302656,  1052932608,   766838016,  419945472,  174553056,   55195776,  13220928,   2370816,   311472,   28896,   1776,    64,    1;
  9: 100776960, 846526464, 3280290048, 7779981312, 12637430784, 14908271616, 13219137792, 8997143040, 4759751808, 1970002944, 638883936, 161834112, 31752000, 4753728, 530064, 42336, 2268, 72, 1;
%F Formula
The row polynomials satisfy (6*n+(2*n+4)*x+x^2)*P(n,x) = (36*n+36+(48*n+72)*x+(18*n+48)*x^2+(2*n+12)*x^3+x^4)*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[Together[((36*n + 36 + (48*n + 72)*x + (18*n + 48)*x^2 + (2*n + 12)*x^3 + x^4)*P[n - 1, x])/(6*n + (2*n + 4)*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] = ((36*n + 36 + (48*n + 72)*x + (18*n + 48)*x^2 + (2*n + 12)*x^3 + x^4)*v[i - 1])/(6*n + (2*n + 4)*x + x^2)); 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

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.