Real-rooted atlas

A330798 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(6 + 5\,x + x^{2} - 21\,n - 18\,n x - n x^{2} + n^{2} x - n^{2} x^{2} + 27\,n^{3} + 24\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(3\,x + 4\,x^{2} + x^{3} - 6\,n x - 6\,n x^{2} - 9\,n^{2} x - 10\,n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2915630
3481128420264
4275825900420702490
51638600685805940198025224396
69996433167644070070350359009924244790
7620163100806511687425604950401921924004034322496528
83893672206413537213673256406104700317444410024561750321287025763058
92466750156227504326300068643960686439604476780018992400503880075582048620268243860
10157378651101650553418915506186609007217710505663126703020334241078690802462229032332301847562812481870
111009756807741468802661129900540810270072108036006614047440425188764019122246005883768001176753601372879270543229653804824
126508724045423936700204559898404612625160069189377400726488462705468192730029684474820115155290253107364975552420440579462002704156314097641130
1342116280083790465207215560857166438551673160064252788600076001869944065570240736041726516832019559304765066599783250159839479802555427420243374040104006003339741725404
142734149780026430114540011732392308003167745923160580753419246076414923585007434965538000542870499600029857877478001229442013800372558186000804386992501167658537510179587254011660035626286189670
Technical details
Node id
n-1b1ae9a68a1f347c · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-da93e8b7035ccffa

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:       2,        2;
  2:       9,       15,        6;
  3:      48,      112,       84,       20;
  4:     275,      825,      900,      420,       70;
  5:    1638,     6006,     8580,     5940,     1980,      252;
  6:    9996,    43316,    76440,    70070,    35035,     9009,      924;
  7:   62016,   310080,   651168,   742560,   495040,   192192,    40040,    3432;
  8:  389367,  2206413,  5372136,  7325640,  6104700,  3174444,  1002456,  175032,  12870;
  9: 2466750, 15622750, 43263000, 68643960, 68643960, 44767800, 18992400, 5038800, 755820, 48620;
%F Formula
The row polynomials satisfy (4*n^3+2*n^2)*P(n,x) = (27*n^3-21*n+6+(24*n^3+n^2-18*n+5)*x+(n^3-n^2-n+1)*x^2)*P(n-1,x) - ((9*n^2+6*n-3)*x+(10*n^2+6*n-4)*x^2+(n^2-1)*x^3)*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[((27*n^3 - 21*n + 6 + (24*n^3 + n^2 - 18*n + 5)*x + (n^3 - n^2 - n + 1)*x^2)*P[n - 1, x] - ((9*n^2 + 6*n - 3)*x + (10*n^2 + 6*n - 4)*x^2 + (n^2 - 1)*x^3)*D[P[n - 1, x], x])/(4*n^3 + 2*n^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] = ((27*n^3 - 21*n + 6 + (24*n^3 + n^2 - 18*n + 5)*x + (n^3 - n^2 - n + 1)*x^2)*v[i - 1] - ((9*n^2 + 6*n - 3)*x + (10*n^2 + 6*n - 4)*x^2 + (n^2 - 1)*x^3)*deriv(v[i - 1], x))/(4*n^3 + 2*n^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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.