Real-rooted atlas

A108838 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 4\,n + n^{2}\right)\,P_{n}(x) = \left(6 + 3\,x + 10\,n + 6\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
022
1538
21416434
34270355152
4132288216646706
54291155115552510573376
614304576572035201100160816514
7486218018270272102190092079231982256
816796707201237601164806370020384364032010415810
958786277134554268612612408408168168420426006429112127864
10208012108528024418803100800244188012337923998408064094505601211002146
11742900424938810623470152423701385670083140203325608875160145860143007151357390504
122674440166409604576264073220224755083565252755225069968818611217907122513282094489614301654146
1396948456518947519556842534526277539838012531551705917528725568590665187065453464175415701298351105151596108000
14353576702554876808303349601603405440204720516018197379201154833680527923968173225052401681286390384663936414961344168494776834
Technical details
Node id
n-8cac8e8480894b25 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5cd009bfc7bc0bb1

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:     2;
  1:     5,      3;
  2:    14,     16,      4;
  3:    42,     70,     35,      5;
  4:   132,    288,    216,     64,      6;
  5:   429,   1155,   1155,    525,    105,      7;
  6:  1430,   4576,   5720,   3520,   1100,    160,     8;
  7:  4862,  18018,  27027,  21021,   9009,   2079,   231,    9;
  8: 16796,  70720, 123760, 116480,  63700,  20384,  3640,  320,  10;
  9: 58786, 277134, 554268, 612612, 408408, 168168, 42042, 6006, 429, 11;
%F Formula
The row polynomials satisfy (n^2+4*n+3)*P(n,x) = (4*n^2+10*n+6+(3*n^2+6*n+3)*x)*P(n-1,x) - ((2*n+3)*x+(2*n+3)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 10*n + 6 + (3*n^2 + 6*n + 3)*x)*P[n - 1, x] - ((2*n + 3)*x + (2*n + 3)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 10*n + 6 + (3*n^2 + 6*n + 3)*x)*v[i - 1] - ((2*n + 3)*x + (2*n + 3)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n + 3)); 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

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