Real-rooted atlas

A114655: 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 Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.

A114655 nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0) · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A114655
\(\left(1 + n\right)\,P_{n}(x) = \left(2 - 3\,x + 2\,n + 3\,n x\right)\,P_{n-1}(x) + \left(4\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
122
2426
3812222
4164824290
532160160402394
6644808004006021806
71281344336028008408428558
82563584125441568078401568112241586
951292164300875264564481881626881442206098
1010242304013824032256033868816934440320432018021037718
1120485632042240012672001774080124185644352079200660022025293446
1240961351681239040464640083635207805952390297610454401452009680264227297738
138192319488351436816107520362419204349030428993536108725762265120251680137283122142078746
141638474547296911365330124814657843221986764818845798494228992274834564580576416416189283642745387038
1532768172032026091520169594880559663104102604902410993382407067174402748345606412806487447366624802548042023937603038

Generated b-files: A114655 ?

Computed in your browser ?

Technical details
Node id
n-6a30765140629bd5 · revision 1 · JSON
Definition source
oeis_catalog:A114655
Fingerprint
f6-9e57d05d9d9a1ce7

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 -1)
  • 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

OEIS matches ?

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.

The atlas's own rows: n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    2;
   2:    4,     2;
   3:    8,    12,      2;
   4:   16,    48,     24,      2;
   5:   32,   160,    160,     40,      2;
   6:   64,   480,    800,    400,     60,      2;
   7:  128,  1344,   3360,   2800,    840,     84,     2;
   8:  256,  3584,  12544,  15680,   7840,   1568,   112,    2;
   9:  512,  9216,  43008,  75264,  56448,  18816,  2688,  144,   2;
  10: 1024, 23040, 138240, 322560, 338688, 169344, 40320, 4320, 180, 2;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (2*n+2+(3*n-3)*x)*P(n-1,x) + (4*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (3*n - 3)*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((2*n + 2 + (3*n - 3)*x)*v[i - 1] + (4*x - 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)

Generated with polytool c48629374597.

A114655: triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1:    2;
   2:    4,     2;
   3:    8,    12,      2;
   4:   16,    48,     24,      2;
   5:   32,   160,    160,     40,      2;
   6:   64,   480,    800,    400,     60,      2;
   7:  128,  1344,   3360,   2800,    840,     84,     2;
   8:  256,  3584,  12544,  15680,   7840,   1568,   112,    2;
   9:  512,  9216,  43008,  75264,  56448,  18816,  2688,  144,   2;
  10: 1024, 23040, 138240, 322560, 338688, 169344, 40320, 4320, 180, 2;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (2*n+2+(3*n-3)*x)*P(n-1,x) + (4*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (3*n - 3)*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((2*n + 2 + (3*n - 3)*x)*v[i - 1] + (4*x - 2*x^2)*deriv(v[i - 1], x))/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-784ea0037cdd0731
  • 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-784ea0037cdd0731
  • 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-784ea0037cdd0731
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-784ea0037cdd0731

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

Used in 9 proof steps

Same fingerprint (5)

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

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