Real-rooted atlas

A257615 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(-35 - 120\,x + 73\,x^{2} - 30\,x^{3} - 8\,x^{4} - 70\,n + 20\,n x - 90\,n x^{2} + 8\,n x^{3} - 16\,n x^{4} - 40\,n^{2} x + 4\,n^{2} x^{2} - 24\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-10\,x - 28\,x^{2} + 42\,x^{3} - 20\,x^{4} + 16\,x^{5} - 20\,n x + 20\,n x^{2} - 24\,n x^{3} + 24\,n x^{4} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 48\,x - 6\,x^{2} + 8\,x^{3} + 24\,n - 4\,n x + 20\,n x^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 10\,x^{2} - 8\,x^{3} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 10 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110515
29512025240
3840214510901254200
470053212032010888062580640
5559104248057440403971506841031251693440
64319955136960146369551353340044421455101201562538707200
732583405815662525458297037849127521313928046569495372709078125958003200
82413650562666088040364691409117622400819317403030537488804676786602687088039062525546752000
91763774106503002725596237497601958640937402655664103401571799894704096984472045596835001920024101953125730637107200
101275657495655489178008335600644453848462845680753449109063066998170444802765307452250524271620880435345397951363548120976562522317642547200
1191542118406458836431051115890851277070473313035945192405476252280245966734451970152142766427820456230033319306479008628280409442966845964249309048828125725323382784000
12653001860056251329565640144307699922910121978354925432045164562823326158012829059085360710728515273378031900797114566407168216965383557800288865556038087532593106798573288024414062524995759652864000
1346369309891059651960528645181494019586676020170951454192810990268239453887502369762655188316752908662567265780801869708658820071806287220465698169010845110581899275920657335330840351444128158504783415364101220703125910559815925760000
143281252082995562803940718640223157386352600953213427117531109202053297613522327775647565262132916276010684751864154559995948920534341795656045680857081830065851179953120870920000159297918962022045107024974993024403223413274514453360597786120610351562534965496931549184000
Technical details
Node id
n-e3bc7bf65fd31062 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-fc62581a8b3dd254

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:        10,          5;
  2:        95,        120,          25;
  3:       840,       2145,        1090,          125;
  4:      7005,      32120,       32010,         8880,          625;
  5:     55910,     424805,      744040,       397150,        68410,         3125;
  6:    431995,    5136960,    14636955,     13533400,      4442145,       510120,       15625;
  7:   3258340,   58156625,   254582970,    378491275,    213139280,     46569495,     3727090,      78125;
  8:  24136505,  626660880,  4036469140,   9117622400,   8193174030,   3053748880,   467678660,   26870880,    390625;
  9: 176377410, 6503002725, 59623749760, 195864093740, 265566410340, 157179989470, 40969844720, 4559683500, 192002410, 1953125;
%F Formula
The row polynomials satisfy (2*n-1+4*x)*P(n,x) = -(70*n+35+(40*n^2-20*n+120)*x+(8*n^3-4*n^2+90*n-73)*x^2+(24*n^2-8*n+30)*x^3+(16*n+8)*x^4)*P(n-2,x) - ((20*n+10)*x+(8*n^2-20*n+28)*x^2-(8*n^2-24*n+42)*x^3-(24*n-20)*x^4-16*x^5)*d/dx P(n-2,x) + (24*n-2+(8*n^2-4*n+48)*x+(20*n-6)*x^2+8*x^3)*P(n-1,x) + ((4*n-2)*x-(4*n-10)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 5*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 10 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(70*n + 35 + (40*n^2 - 20*n + 120)*x + (8*n^3 - 4*n^2 + 90*n - 73)*x^2 + (24*n^2 - 8*n + 30)*x^3 + (16*n + 8)*x^4)*P[n - 2, x] - ((20*n + 10)*x + (8*n^2 - 20*n + 28)*x^2 - (8*n^2 - 24*n + 42)*x^3 - (24*n - 20)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (24*n - 2 + (8*n^2 - 4*n + 48)*x + (20*n - 6)*x^2 + 8*x^3)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 10)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(2*n - 1 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 10 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(70*n + 35 + (40*n^2 - 20*n + 120)*x + (8*n^3 - 4*n^2 + 90*n - 73)*x^2 + (24*n^2 - 8*n + 30)*x^3 + (16*n + 8)*x^4)*v[i - 2] - ((20*n + 10)*x + (8*n^2 - 20*n + 28)*x^2 - (8*n^2 - 24*n + 42)*x^3 - (24*n - 20)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (24*n - 2 + (8*n^2 - 4*n + 48)*x + (20*n - 6)*x^2 + 8*x^3)*v[i - 1] + ((4*n - 2)*x - (4*n - 10)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(2*n - 1 + 4*x)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.