Real-rooted atlas

A069466 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-2\,n - 8\,n x - 10\,n x^{2} - 4\,n x^{3} + 4\,n^{2} + 16\,n^{2} x + 20\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x - 12\,x^{2} - 12\,x^{3} - 4\,x^{4} + 8\,n x + 24\,n x^{2} + 24\,n x^{3} + 8\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
26363678
3202406004001260
470140063009800490022470
525275605292014112015876063504424116
6924388083880801552320291060025613288537768305836
7343219219225945921441440039639600570810244122518411778624166929048
8128709266401621620011891880044594550092756664010821610806625476001656369003419932230
948620437580096267600898497600438017580012264492240204408204002002366080010637569800236390440071109813060
1018475620323160548725320634082592038837558760139815211536310700470080431176162560363804887160170673897680341347795361496053026468
117054329311702430263032804236824592031776184440014235730629124033457011584740839042944087974636349606516639729600273698868643249763430662431777397077608
1227041564218483361624116093627068601560024361741404001325278732377646384755633216107915553922176168618053003400174863166077600115409689611216438747580340167312459672336680354749147164
13104006001892909200851809140001665760096000176987010200001146875826096004842364599072001383532742592000272383008697800036990285131800003403106232125600202498883233920070312112234000010817248036000014664155597771400
1440116600842448600043807327200099296608320001228795527960009338846012496004669423006248000160094503071360003852273980154600065631334476708000787576013720496006508892675376000035256501991620000112653911689200001609341595560000317877850826299800
Technical details
Node id
n-8f0a36c9d23410b3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-fd15deff59829444

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,       4;
  2:     6,      36,       36;
  3:    20,     240,      600,       400;
  4:    70,    1400,     6300,      9800,       4900;
  5:   252,    7560,    52920,    141120,     158760,       63504;
  6:   924,   38808,   388080,   1552320,    2910600,     2561328,      853776;
  7:  3432,  192192,  2594592,  14414400,   39639600,    57081024,    41225184,    11778624;
  8: 12870,  926640, 16216200, 118918800,  445945500,   927566640,  1082161080,   662547600,   165636900;
  9: 48620, 4375800, 96267600, 898497600, 4380175800, 12264492240, 20440820400, 20023660800, 10637569800, 2363904400;
%F Formula
The row polynomials satisfy (n^2+2*n^2*x+n^2*x^2)*P(n,x) = (4*n^2-2*n+(16*n^2-8*n)*x+(20*n^2-10*n)*x^2+(8*n^2-4*n)*x^3)*P(n-1,x) + ((8*n-4)*x+(24*n-12)*x^2+(24*n-12)*x^3+(8*n-4)*x^4)*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[((4*n^2 - 2*n + (16*n^2 - 8*n)*x + (20*n^2 - 10*n)*x^2 + (8*n^2 - 4*n)*x^3)*P[n - 1, x] + ((8*n - 4)*x + (24*n - 12)*x^2 + (24*n - 12)*x^3 + (8*n - 4)*x^4)*D[P[n - 1, x], x])/(n^2 + 2*n^2*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] = ((4*n^2 - 2*n + (16*n^2 - 8*n)*x + (20*n^2 - 10*n)*x^2 + (8*n^2 - 4*n)*x^3)*v[i - 1] + ((8*n - 4)*x + (24*n - 12)*x^2 + (24*n - 12)*x^3 + (8*n - 4)*x^4)*deriv(v[i - 1], x))/(n^2 + 2*n^2*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

Candidate equalities ?

Same fingerprint ?

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