A114655 → x ↦ x+1

OEIS matches ?

• A151374: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A052701: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:       2;
   2:       6,       2;
   3:      22,      16,       2;
   4:      90,     102,      30,       2;
   5:     394,     608,     292,      48,       2;
   6:    1806,    3530,    2380,     660,      70,      2;
   7:    8558,   20256,   17670,    7040,    1290,     96,     2;
   8:   41586,  115598,  124026,   65030,   17430,   2282,   126,    2;
   9:  206098,  658048,  839048,  548128,  196028,  38080,  3752,  160,   2;
  10: 1037718, 3742290, 5532120, 4338600, 1954260, 512316, 75768, 5832, 198, 2;

The row polynomials satisfy (n+1)*P(n,x) = (5*n-1+(3*n-3)*x)*P(n-1,x) + (2-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.

P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 1 + (3*n - 3)*x)*P[n - 1, x] + (2 - 2*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] = 2; for(n = 2, N, my(i = n); v[i] = ((5*n - 1 + (3*n - 3)*x)*v[i - 1] + (2 - 2*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 (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-5ebd2cd7e542e718
• 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-5ebd2cd7e542e718
• 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-5ebd2cd7e542e718
• 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-5ebd2cd7e542e718

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-5ebd2cd7e542e718

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-784ea0037cdd0731 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

Same fingerprint (2)

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

• A114687 → x ↦ x/(x+1)
• A114656 → basis A007318

References

1. [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