Real-rooted atlas

A081579 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(30\,x^{2} + 40\,x^{3} + n - 4\,n x - 50\,n x^{2} - 60\,n x^{3} + 6\,n^{2} x + 20\,n^{2} x^{2} + 20\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(31\,x^{2} + 110\,x^{3} + 80\,x^{4} - n x - 22\,n x^{2} - 60\,n x^{3} - 40\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{2} + 21\,x^{3} + 40\,x^{4} + 20\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
218817
3114362475
412010016080361
51262005606402561683
61323361440280024968327937
713850830168720128969408268837275
81447165504218564729656384348168704175321
915096091204696013795223744023680012672028160824163
10156124014080900803413127899521126400965120455680911363875057
111621556206001585607457922212224420787251148803840000162201629491218218475
121681908288962610401479552542937613157376211960322243584014985216572620895436885655881
1317422963918440745627182721200588835848192731911681021189129567641657544704200744963088384402715443
14180272051680609040469292824404800874956802192345603859148804744417283985817602180096006995968099942401893393377
Technical details
Node id
n-6753ec6d7a3a663e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-076e7110b92525fe

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: 1,  2;
  2: 1,  8,   8;
  3: 1, 14,  36,   24;
  4: 1, 20, 100,  160,    80;
  5: 1, 26, 200,  560,   640,    256;
  6: 1, 32, 336, 1440,  2800,   2496,    832;
  7: 1, 38, 508, 3016,  8720,  12896,   9408,   2688;
  8: 1, 44, 716, 5504, 21856,  47296,  56384,  34816,   8704;
  9: 1, 50, 960, 9120, 46960, 137952, 237440, 236800, 126720, 28160;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+(6*n^2-4*n)*x+(20*n^2-50*n+30)*x^2+(20*n^2-60*n+40)*x^3)*P(n-1,x) - (n*x+(22*n-31)*x^2+(60*n-110)*x^3+(40*n-80)*x^4)*d/dx P(n-1,x) + (x^2+21*x^3+40*x^4+20*x^5)*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[((n + (6*n^2 - 4*n)*x + (20*n^2 - 50*n + 30)*x^2 + (20*n^2 - 60*n + 40)*x^3)*P[n - 1, x] - (n*x + (22*n - 31)*x^2 + (60*n - 110)*x^3 + (40*n - 80)*x^4)*D[P[n - 1, x], x] + (x^2 + 21*x^3 + 40*x^4 + 20*x^5)*D[P[n - 1, x], {x, 2}])/(n)]];
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] = ((n + (6*n^2 - 4*n)*x + (20*n^2 - 50*n + 30)*x^2 + (20*n^2 - 60*n + 40)*x^3)*v[i - 1] - (n*x + (22*n - 31)*x^2 + (60*n - 110)*x^3 + (40*n - 80)*x^4)*deriv(v[i - 1], x) + (x^2 + 21*x^3 + 40*x^4 + 20*x^5)*deriv(deriv(v[i - 1], x), x))/(n)); 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 ?

Same fingerprint ?

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