Real-rooted atlas

A126216 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-2 - 16\,x - 38\,x^{2} - 36\,x^{3} - 12\,x^{4} + 11\,n + 46\,n x + 71\,n x^{2} + 48\,n x^{3} + 12\,n x^{4}\right)\,P_{n-1}(x) + \left(-4 - 26\,x - 66\,x^{2} - 82\,x^{3} - 50\,x^{4} - 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
211291959
345177231100553
41971030201217405625541
59035890153201986012830330458107
64279334471086451877051819059374720071629799
7207931894277378771593025205852515920596823111249966999013
81030491072076486959212612740203739342101694013519912495850479377479320521
951885960690243149222495145204184438674237890772204149568112397112360234185120632913245487
1026467233438118520063977569267333015665920202425274544260270808019118023809198721502617904103346310210651843699
111364886919495043712637519954907647470126854547602291601127229521341696271198402201741037661074387498901903685366221060008125596518593
1271039373110653720278885841323403488792098973968940204368616144307244988336338847223176272071800990155105037780595921531921385406304814738303081494632730541
13372693519628714764648886902792231992234760749613783204174157236739229930352085923852443519496371370764879426480364773721357513746160473802802616100908752412990418619217928077470947
141968801519357584538993011677558891559000330979554130101424914306171390511276655286557654070910931616145802642830945392115691277312514300210522111709255101999374386397030173562681399967015401391216532981070559
Technical details
Node id
n-6d17f43e8a2b3f05 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-58bc7a5101bba6c3

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,       4;
  2:     11,      29,       19;
  3:     45,     177,      231,      100;
  4:    197,    1030,     2012,     1740,       562;
  5:    903,    5890,    15320,    19860,     12830,      3304;
  6:   4279,   33447,   108645,   187705,    181905,     93747,     20071;
  7:  20793,  189427,   737877,  1593025,   2058525,   1592059,    682311,    124996;
  8: 103049, 1072076,  4869592, 12612740,  20373934,  21016940,  13519912,   4958504,   793774;
  9: 518859, 6069024, 31492224, 95145204, 184438674, 237890772, 204149568, 112397112, 36023418, 5120632;
%F Formula
The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (11*n-2+(46*n-16)*x+(71*n-38)*x^2+(48*n-36)*x^3+(12*n-12)*x^4)*P(n-1,x) - (4+26*x+66*x^2+82*x^3+50*x^4+12*x^5)*d/dx 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[((11*n - 2 + (46*n - 16)*x + (71*n - 38)*x^2 + (48*n - 36)*x^3 + (12*n - 12)*x^4)*P[n - 1, x] - (4 + 26*x + 66*x^2 + 82*x^3 + 50*x^4 + 12*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (n + 2)*x^2)]];
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] = ((11*n - 2 + (46*n - 16)*x + (71*n - 38)*x^2 + (48*n - 36)*x^3 + (12*n - 12)*x^4)*v[i - 1] - (4 + 26*x + 66*x^2 + 82*x^3 + 50*x^4 + 12*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (n + 2)*x^2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023