Real-rooted atlas

A116412 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 4\,n\right)\,P_{n}(x) = \left(96 + 32\,x - 60\,n - 16\,n x + 12\,n^{2}\right)\,P_{n-1}(x) + \left(274 + 232\,x + 40\,x^{2} - 74\,n - 56\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x) + \left(111 + 158\,x + 68\,x^{2} + 8\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1628
23324461
3183204728467
410141524828192163574
5561910626781227844803227353
6311377096865844317768400115264209341
71725424602305158443145441125602361626881281602152
8956121292141238391482831424128064036345663168614425612261769
9529823118245076275065562381073613032816467452810980481628161382451293843143
102935951811249832019138320019023936012229032053089920157557123154944407808307201024718210846
111626922836865423261300922460146040852010786502405501194561976016004997068887160329984006758420485496691637
129015399694154310192867684319210857372000906141792053117481602241693888686032896151054848233318402399232147456409642067895689
134995776694249596424425696868837678623616864731786013604848923548823525009472845866752022534295044391961606085939256770563194888192321958728008
142768350337714905577611236910187672455698126345657203923828842286146048023195209632095884843008300097128967074945024123655065615534489613258752688128163842464050574501
Technical details
Node id
n-9106f6ba15ffc09e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-20bdec796f87d0c7

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:       6,        2;
  2:      33,       24,        4;
  3:     183,      204,       72,        8;
  4:    1014,     1524,      828,      192,       16;
  5:    5619,    10626,     7812,     2784,      480,      32;
  6:   31137,    70968,    65844,    31776,     8400,    1152,      64;
  7:  172542,   460230,   515844,   314544,   112560,   23616,    2688,    128;
  8:  956121,  2921412,  3839148,  2831424,  1280640,  363456,   63168,   6144,   256;
  9: 5298231, 18245076, 27506556, 23810736, 13032816, 4674528, 1098048, 162816, 13824, 512;
%F Formula
The row polynomials satisfy (4*n+4)*P(n,x) = (12*n^2-60*n+96-(16*n-32)*x)*P(n-1,x) - (74*n-274+(56*n-232)*x+(8*n-40)*x^2)*d/dx P(n-1,x) + (111+158*x+68*x^2+8*x^3)*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[((12*n^2 - 60*n + 96 - (16*n - 32)*x)*P[n - 1, x] - (74*n - 274 + (56*n - 232)*x + (8*n - 40)*x^2)*D[P[n - 1, x], x] + (111 + 158*x + 68*x^2 + 8*x^3)*D[P[n - 1, x], {x, 2}])/(4*n + 4)]];
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] = ((12*n^2 - 60*n + 96 - (16*n - 32)*x)*v[i - 1] - (74*n - 274 + (56*n - 232)*x + (8*n - 40)*x^2)*deriv(v[i - 1], x) + (111 + 158*x + 68*x^2 + 8*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n + 4)); 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

Candidate equalities ?

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