A053117 → x ↦ x+1

OEIS matches ?

• A010905: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A195503: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A001353: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
   1:  1;
   2:  2,   2;
   3:  3,   8,    4;
   4:  4,  20,   24,     8;
   5:  5,  40,   84,    64,    16;
   6:  6,  70,  224,   288,   160,    32;
   7:  7, 112,  504,   960,   880,   384,    64;
   8:  8, 168, 1008,  2640,  3520,  2496,   896,   128;
   9:  9, 240, 1848,  6336, 11440, 11648,  6720,  2048,  256;
  10: 10, 330, 3168, 13728, 32032, 43680, 35840, 17408, 4608, 512;

The row polynomials satisfy (n-1)*P(n,x) = (n+n*x)*P(n-1,x) + (2*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + n*x)*P[n - 1, x] + (2*x + x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n + n*x)*v[i - 1] + (2*x + x^2)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-2, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-5342624c2722e8d8
• Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-5342624c2722e8d8
• Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-5342624c2722e8d8
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-5342624c2722e8d8

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image affine, shift 0 e-5342624c2722e8d8

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 1)
    • theorem_check three_term_recurrence e-dc563681f97c05c6 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

• chebyshev_u → x ↦ x+1
• A091044 → basis A013609

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023