Real-rooted atlas

A272866 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1510318
225100105405275
3125750135011004509073872
4625500012750160001137548001190160951909
53125312501031251750001781251162504987514000247525011673486
6156251875007593751625000215625019125001176875510000155250325004455360138535703
7781251093750525000013562500222031252493750020059375118250005158125166250039077565100728049015106293660

All 15 rows

Technical details
Node id
n-5bd8ef4eded0ab6c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6a35350bf0e37fe8

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:       5,       10,         3;
  2:      25,      100,       105,        40,          5;
  3:     125,      750,      1350,      1100,        450,         90,          7;
  4:     625,     5000,     12750,     16000,      11375,       4800,       1190,        160,          9;
  5:    3125,    31250,    103125,    175000,     178125,     116250,      49875,      14000,       2475,        250,        11;
  6:   15625,   187500,    759375,   1625000,    2156250,    1912500,    1176875,     510000,     155250,      32500,      4455,       360,       13;
  7:   78125,  1093750,   5250000,  13562500,   22203125,   24937500,   20059375,   11825000,    5158125,    1662500,    390775,     65100,     7280,      490,      15;
  8:  390625,  6250000,  34687500, 105000000,  204531250,  278250000,  276850000,  207000000,  118096875,   51750000,  17402000,   4452000,   850850,   117600,   11100,    640,    17;
  9: 1953125, 35156250, 221484375, 768750000, 1736718750, 2775937500, 3293718750, 2988000000, 2110809375, 1173593750, 515975625, 179280000, 48935250, 10363500, 1667250, 196800, 16065, 810, 19;
%F Formula
The row polynomials satisfy (5+5*n*x+(2*n-1)*x^2)*P(n,x) = (25+(25*n+50)*x+(35*n+35)*x^2+(15*n+10)*x^3+(2*n+1)*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 + (25*n + 50)*x + (35*n + 35)*x^2 + (15*n + 10)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(5 + 5*n*x + (2*n - 1)*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 + (25*n + 50)*x + (35*n + 35)*x^2 + (15*n + 10)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(5 + 5*n*x + (2*n - 1)*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.