Real-rooted atlas

A272866 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3\,x + x^{2} + 5\,n + 2\,n x\right)\,P_{n}(x) = \left(25 + 50\,x + 35\,x^{2} + 10\,x^{3} + x^{4} + 25\,n + 35\,n x + 15\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1107118
27512065141275
3500127512756351652113872
431251100016250131006275182031028151909
5187508437516562518625013250062125193753950500351673486
610937560000014718752131250202500013275006143752017504650073007354218535703
7625000404687511921875211531252522187521350000132125006063125206937552150095200121451015491106293660

All 15 rows

Technical details
Node id
n-6912aa82d26738f3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3ee923700ed89645

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:       10,         7,         1;
  2:       75,       120,        65,         14,          1;
  3:      500,      1275,      1275,        635,        165,         21,          1;
  4:     3125,     11000,     16250,      13100,       6275,       1820,        310,         28,          1;
  5:    18750,     84375,    165625,     186250,     132500,      62125,      19375,       3950,        500,        35,         1;
  6:   109375,    600000,   1471875,    2131250,    2025000,    1327500,     614375,     201750,      46500,      7300,       735,       42,        1;
  7:   625000,   4046875,  11921875,   21153125,   25221875,   21350000,   13212500,    6063125,    2069375,    521500,     95200,    12145,     1015,      49,      1;
  8:  3515625,  26250000,  90312500,  189875000,  272781250,  283675000,  220675000,  130850000,   59696875,  20995000,   5663000,  1156400,   174650,   18760,   1340,    56,    1;
  9: 19531250, 165234375, 650390625, 1581562500, 2660625000, 3285843750, 3085031250, 2249775000, 1290768750, 586428125, 211246875, 60111000, 13387500, 2296350, 295650, 27420, 1710, 63, 1;
%F Formula
The row polynomials satisfy (5*n+(2*n+3)*x+x^2)*P(n,x) = (25*n+25+(35*n+50)*x+(15*n+35)*x^2+(2*n+10)*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[((25*n + 25 + (35*n + 50)*x + (15*n + 35)*x^2 + (2*n + 10)*x^3 + x^4)*P[n - 1, x])/(5*n + (2*n + 3)*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] = ((25*n + 25 + (35*n + 50)*x + (15*n + 35)*x^2 + (2*n + 10)*x^3 + x^4)*v[i - 1])/(5*n + (2*n + 3)*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.