Real-rooted atlas

A113413 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(31 + 201\,x + 534\,x^{2} + 748\,x^{3} + 585\,x^{4} + 243\,x^{5} + 42\,x^{6} - 33\,n - 229\,n x - 651\,n x^{2} - 972\,n x^{3} - 805\,n x^{4} - 351\,n x^{5} - 63\,n x^{6} + 8\,n^{2} + 56\,n^{2} x + 165\,n^{2} x^{2} + 260\,n^{2} x^{3} + 230\,n^{2} x^{4} + 108\,n^{2} x^{5} + 21\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(22 + 203\,x + 771\,x^{2} + 1578\,x^{3} + 1892\,x^{4} + 1335\,x^{5} + 515\,x^{6} + 84\,x^{7} - 8\,n - 76\,n x - 300\,n x^{2} - 642\,n x^{3} - 808\,n x^{4} - 600\,n x^{5} - 244\,n x^{6} - 42\,n x^{7}\right)\,P_{n-1}'(x) + \left(2 + 24\,x + 121\,x^{2} + 336\,x^{3} + 565\,x^{4} + 592\,x^{5} + 379\,x^{6} + 136\,x^{7} + 21\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1358
27262356
31797179107400
44131689411004972848
599961367068906359230920288
623927981346534084478573534210727144512
7577790945933146481276979310527191135498351029376
813932188014887657303213644702057992192004410131442315217332352
9336359549464572209550860217701143175014336236114524205288363107558952229120
108119160002140791572834802452879856185548886448949511708066430635272701464996919372033536
111960142548141686952433106393989034252291642480160526647906350607436341376849133139241867232144432650034176
12473211121844121115947873205234318878310566906002356461100383470810445193317193761787556209924967470519446810784852918876473344
13114243293685734639986248192766120512514541873772991071097879620420683988290170391733035121553522714596994115181169423547058303501037445134459113472
142758077641814977683337654894364097473127158608970664578425663710011590746416664436173321011382650619753845118313428217769262392959797177372856222327695367957766467584
Technical details
Node id
n-4c0f506cbc379896 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3c9edfa663579ff7

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:    3,     5;
  2:    7,    26,     23;
  3:   17,    97,    179,     107;
  4:   41,   316,    894,    1100,     497;
  5:   99,   961,   3670,    6890,    6359,     2309;
  6:  239,  2798,  13465,   34084,   47857,    35342,    10727;
  7:  577,  7909,  45933,  146481,  276979,   310527,   191135,    49835;
  8: 1393, 21880, 148876,  573032, 1364470,  2057992,  1920044,  1013144,  231521;
  9: 3363, 59549, 464572, 2095508, 6021770, 11431750, 14336236, 11452420, 5288363, 1075589;
%F Formula
The row polynomials satisfy (n+1+(3*n+3)*x+(3*n+3)*x^2+(n+1)*x^3)*P(n,x) = (8*n^2-33*n+31+(56*n^2-229*n+201)*x+(165*n^2-651*n+534)*x^2+(260*n^2-972*n+748)*x^3+(230*n^2-805*n+585)*x^4+(108*n^2-351*n+243)*x^5+(21*n^2-63*n+42)*x^6)*P(n-1,x) - (8*n-22+(76*n-203)*x+(300*n-771)*x^2+(642*n-1578)*x^3+(808*n-1892)*x^4+(600*n-1335)*x^5+(244*n-515)*x^6+(42*n-84)*x^7)*d/dx P(n-1,x) + (2+24*x+121*x^2+336*x^3+565*x^4+592*x^5+379*x^6+136*x^7+21*x^8)*d^2/dx^2 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[((8*n^2 - 33*n + 31 + (56*n^2 - 229*n + 201)*x + (165*n^2 - 651*n + 534)*x^2 + (260*n^2 - 972*n + 748)*x^3 + (230*n^2 - 805*n + 585)*x^4 + (108*n^2 - 351*n + 243)*x^5 + (21*n^2 - 63*n + 42)*x^6)*P[n - 1, x] - (8*n - 22 + (76*n - 203)*x + (300*n - 771)*x^2 + (642*n - 1578)*x^3 + (808*n - 1892)*x^4 + (600*n - 1335)*x^5 + (244*n - 515)*x^6 + (42*n - 84)*x^7)*D[P[n - 1, x], x] + (2 + 24*x + 121*x^2 + 336*x^3 + 565*x^4 + 592*x^5 + 379*x^6 + 136*x^7 + 21*x^8)*D[P[n - 1, x], {x, 2}])/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*x^3)]];
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] = ((8*n^2 - 33*n + 31 + (56*n^2 - 229*n + 201)*x + (165*n^2 - 651*n + 534)*x^2 + (260*n^2 - 972*n + 748)*x^3 + (230*n^2 - 805*n + 585)*x^4 + (108*n^2 - 351*n + 243)*x^5 + (21*n^2 - 63*n + 42)*x^6)*v[i - 1] - (8*n - 22 + (76*n - 203)*x + (300*n - 771)*x^2 + (642*n - 1578)*x^3 + (808*n - 1892)*x^4 + (600*n - 1335)*x^5 + (244*n - 515)*x^6 + (42*n - 84)*x^7)*deriv(v[i - 1], x) + (2 + 24*x + 121*x^2 + 336*x^3 + 565*x^4 + 592*x^5 + 379*x^6 + 136*x^7 + 21*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*x^3)); 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