Real-rooted atlas

A166960 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,x + n\right)\,P_{n}(x) = \left(12 - 43\,x + 60\,x^{2} - 38\,x^{3} + 8\,x^{4} - 4\,n + 15\,n x - 37\,n x^{2} + 36\,n x^{3} - 8\,n x^{4} - 2\,n^{2} x + 10\,n^{2} x^{2} - 6\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(15\,x - 71\,x^{2} + 98\,x^{3} - 46\,x^{4} + 8\,x^{5} - 5\,n x + 29\,n x^{2} - 29\,n x^{3} + 6\,n x^{4} - 4\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(3\,x^{2} - 16\,x^{3} + 22\,x^{4} - 8\,x^{5} - n x^{2} + 7\,n x^{3} - 6\,n x^{4} - n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(-19 + 48\,x - 38\,x^{2} + 8\,x^{3} + 5\,n - 15\,n x + 10\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-20\,x + 24\,x^{2} - 4\,x^{3} + 5\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(-4\,x^{2} + 4\,x^{3} + n x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
278116
32869241122
41136524446211272
5454620980922302146117204
618195815214612774276103293241291028
77280536101257427622729755640804177569215997180
8291254890428441478906688034627538450373832015659814421147382600
9116506443342257406283601897609404126022977628084449022337472555048295814246658240
10466031400522600122269529255216772948054920806230191239281362517326898123148200188721960081141482773728
11186413236116938611996364728001398199913805229817335092012186034400822477252779682039199261063731701262206811212815386987556000
127456537325369349883235717112136367535310151909298659175033573773714832992223206879381362839183264404152295971751313728365220032848243901232097423617360
132982615829297408908952194446492484952284862031306366153204282763042836583591810631885806110551656353234211925956294962334598981064660625095148409240206339160248938111219578835374896
14119304643263739315111283935922343229924416105583905452141121617602583671240344543342818588151448635731733755405749785049598004977319144155369282722339393008705004402391368004328252197956429980601604015037231844352
Technical details
Node id
n-2304b4108bb95699 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3d50e139fbffd7c0

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:      2,        1;
  2:      7,        8,         1;
  3:     28,       69,        24,          1;
  4:    113,      652,       444,         62,          1;
  5:    454,     6209,      8092,       2302,        146,         1;
  6:   1819,    58152,    146127,      74276,      10329,       324,        1;
  7:   7280,   536101,   2574276,    2272975,     564080,     41775,      692,      1;
  8:  29125,  4890428,  44147890,   66880346,   27538450,   3738320,   156598,   1442,    1;
  9: 116506, 44334225, 740628360, 1897609404, 1260229776, 280844490, 22337472, 555048, 2958, 1;
%F Formula
The row polynomials satisfy (n-4+4*x)*P(n,x) = -(4*n-12+(2*n^2-15*n+43)*x+(n^3-10*n^2+37*n-60)*x^2+(6*n^2-36*n+38)*x^3+(8*n-8)*x^4)*P(n-2,x) - ((5*n-15)*x+(4*n^2-29*n+71)*x^2-(n^2-29*n+98)*x^3-(6*n-46)*x^4-8*x^5)*d/dx P(n-2,x) - ((n-3)*x^2+(n^2-7*n+16)*x^3+(6*n-22)*x^4+8*x^5)*d^2/dx^2 P(n-2,x) + (5*n-19+(2*n^2-15*n+48)*x+(10*n-38)*x^2+8*x^3)*P(n-1,x) + ((5*n-20)*x-(n-24)*x^2-4*x^3)*d/dx P(n-1,x) + ((n-4)*x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 12 + (2*n^2 - 15*n + 43)*x + (n^3 - 10*n^2 + 37*n - 60)*x^2 + (6*n^2 - 36*n + 38)*x^3 + (8*n - 8)*x^4)*P[n - 2, x] - ((5*n - 15)*x + (4*n^2 - 29*n + 71)*x^2 - (n^2 - 29*n + 98)*x^3 - (6*n - 46)*x^4 - 8*x^5)*D[P[n - 2, x], x] - ((n - 3)*x^2 + (n^2 - 7*n + 16)*x^3 + (6*n - 22)*x^4 + 8*x^5)*D[P[n - 2, x], {x, 2}] + (5*n - 19 + (2*n^2 - 15*n + 48)*x + (10*n - 38)*x^2 + 8*x^3)*P[n - 1, x] + ((5*n - 20)*x - (n - 24)*x^2 - 4*x^3)*D[P[n - 1, x], x] + ((n - 4)*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}])/(n - 4 + 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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 12 + (2*n^2 - 15*n + 43)*x + (n^3 - 10*n^2 + 37*n - 60)*x^2 + (6*n^2 - 36*n + 38)*x^3 + (8*n - 8)*x^4)*v[i - 2] - ((5*n - 15)*x + (4*n^2 - 29*n + 71)*x^2 - (n^2 - 29*n + 98)*x^3 - (6*n - 46)*x^4 - 8*x^5)*deriv(v[i - 2], x) - ((n - 3)*x^2 + (n^2 - 7*n + 16)*x^3 + (6*n - 22)*x^4 + 8*x^5)*deriv(deriv(v[i - 2], x), x) + (5*n - 19 + (2*n^2 - 15*n + 48)*x + (10*n - 38)*x^2 + 8*x^3)*v[i - 1] + ((5*n - 20)*x - (n - 24)*x^2 - 4*x^3)*deriv(v[i - 1], x) + ((n - 4)*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), x))/(n - 4 + 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. [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.