Real-rooted atlas

A062264 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(4\,n + n^{2}\right)\,P_{n}(x) = \left(12 - 4\,x + 14\,n + 6\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x - 4\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1617
22814143
3120108241253
44956602703611462
52002357522005505018378
680081801815015572099066147818
731824866329172847775127401638841272422
812597040310451979234652812740025480254810411550927
9497420183141027907202279088107956829988047040378012618829033
1019612568171900143896501395360081396002930256642600816005400150150276013
11772616035956360719127208090181056279520250699687162848127908013464074801761286430763
1230421755156454740350574510449454500364058145194164344689424121611640823982752131801009820411632808572
1311975985067464715516749860402412777510223405325013919870255939144641732250523389185842792753260401333823419313861092
1447143560028875430507870883475125934135601313623311093830236504691511825164976240040419178867365298732231548412017290266153163187748
Technical details
Node id
n-54cdd4c005ce1726 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-0e61d18a7a9911bc

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:      6,       1;
  2:     28,      14,       1;
  3:    120,     108,      24,       1;
  4:    495,     660,     270,      36,       1;
  5:   2002,    3575,    2200,     550,      50,      1;
  6:   8008,   18018,   15015,    5720,     990,     66,     1;
  7:  31824,   86632,   91728,   47775,   12740,   1638,    84,    1;
  8: 125970,  403104,  519792,  346528,  127400,  25480,  2548,  104,   1;
  9: 497420, 1831410, 2790720, 2279088, 1079568, 299880, 47040, 3780, 126, 1;
%F Formula
The row polynomials satisfy (n^2+4*n)*P(n,x) = (4*n^2+14*n+12+(3*n^2+6*n-4)*x)*P(n-1,x) - ((2*n+4)*x+(2*n+4)*x^2)*d/dx 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[((4*n^2 + 14*n + 12 + (3*n^2 + 6*n - 4)*x)*P[n - 1, x] - ((2*n + 4)*x + (2*n + 4)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n)]];
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] = ((4*n^2 + 14*n + 12 + (3*n^2 + 6*n - 4)*x)*v[i - 1] - ((2*n + 4)*x + (2*n + 4)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.