Real-rooted atlas

A144436: Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4, read by rows.

A144436 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A144436
\(P_{n}(x) = \left(1 + n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x) + \left(4\,x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
218110
312323148
4154170541280
5111781881811711872
61244325572243255244114224
71499116974844348443116974991121280
8110103956027697451332627697439560101011148416
91203312875614316044422246442224614316041287562033111969280
10140804086056914704332403545512025633240354691470440860540801136255744
1118175127474731851285227245818581092998581092998227245818318512851274747817511682946048
12116366393048614178703014505494715437485948835578299654374859481450549471141787030393048616366122423321600
131327491202055061548245487980227954658884419910431474054810431474054846588844199879802279561548245412020550327491320658286592
1416551636552851262189836051327569089373315491028117125272541917024589807521171252725419373315491028513275690892621898360365528516551614899567585280

Generated b-files: A144436, A144436 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-9189beb5ff99695e · revision 1 · JSON
Definition source
oeis_catalog:A144436
Fingerprint
f6-e7d9ff2a05154d95

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: yes
  • γ-positive (gamma_positive): yes
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,    1;
  2: 1,    8,      1;
  3: 1,   23,     23,       1;
  4: 1,   54,    170,      54,       1;
  5: 1,  117,    818,     818,     117,       1;
  6: 1,  244,   3255,    7224,    3255,     244,       1;
  7: 1,  499,  11697,   48443,   48443,   11697,     499,      1;
  8: 1, 1010,  39560,  276974,  513326,  276974,   39560,   1010,    1;
  9: 1, 2033, 128756, 1431604, 4422246, 4422246, 1431604, 128756, 2033, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+n*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) + 4*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + 4*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + n*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + 4*x*v[i - 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)

Generated with polytool c48629374597.

A144436: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,    1;
   3: 1,    8,      1;
   4: 1,   23,     23,       1;
   5: 1,   54,    170,      54,       1;
   6: 1,  117,    818,     818,     117,       1;
   7: 1,  244,   3255,    7224,    3255,     244,       1;
   8: 1,  499,  11697,   48443,   48443,   11697,     499,      1;
   9: 1, 1010,  39560,  276974,  513326,  276974,   39560,   1010,    1;
  10: 1, 2033, 128756, 1431604, 4422246, 4422246, 1431604, 128756, 2033, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) + 4*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + 4*x*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + 4*x*v[i - 2]); 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.

A144436: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,    1;
   3: 1,    8,      1;
   4: 1,   23,     23,       1;
   5: 1,   54,    170,      54,       1;
   6: 1,  117,    818,     818,     117,       1;
   7: 1,  244,   3255,    7224,    3255,     244,       1;
   8: 1,  499,  11697,   48443,   48443,   11697,     499,      1;
   9: 1, 1010,  39560,  276974,  513326,  276974,   39560,   1010,    1;
  10: 1, 2033, 128756, 1431604, 4422246, 4422246, 1431604, 128756, 2033, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) + 4*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + 4*x*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + 4*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(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 >= 0, 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-f9fe2d240d80ec70
  • Coefficients of one sign. Every nonzero row, n >= 0: 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-f9fe2d240d80ec70
  • Hadamard kernel (Malo). Rows P_n, n >= 0, 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-f9fe2d240d80ec70
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-f9fe2d240d80ec70

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 1 proof step

No candidate equalities or related sequences.

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