Real-rooted atlas

delannoy → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(4 + 36\,x + 120\,x^{2} + 200\,x^{3} + 180\,x^{4} + 84\,x^{5} + 16\,x^{6} - 6\,n - 51\,n x - 169\,n x^{2} - 285\,n x^{3} - 261\,n x^{4} - 124\,n x^{5} - 24\,n x^{6} + 3\,n^{2} + 20\,n^{2} x + 58\,n^{2} x^{2} + 92\,n^{2} x^{3} + 83\,n^{2} x^{4} + 40\,n^{2} x^{5} + 8\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(1 + 25\,x + 142\,x^{2} + 370\,x^{3} + 525\,x^{4} + 421\,x^{5} + 180\,x^{6} + 32\,x^{7} - n - 14\,n x - 70\,n x^{2} - 176\,n x^{3} - 249\,n x^{4} - 202\,n x^{5} - 88\,n x^{6} - 16\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 13\,x^{2} + 58\,x^{3} + 130\,x^{4} + 165\,x^{5} + 121\,x^{6} + 48\,x^{7} + 8\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
215511
318181239
4111405829139
51147116417570495
61171113576015071691763
71201606641550204814284086279

Show all 15 stored rows

Generated b-files: A007482 (row sums), A209695 ?

Computed in your browser ?

Technical details
Node id
n-09ec0cbf57326f93 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f8b12148df73b33e

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

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,  2;
  2: 1,  5,   5;
  3: 1,  8,  18,   12;
  4: 1, 11,  40,   58,   29;
  5: 1, 14,  71,  164,  175,    70;
  6: 1, 17, 111,  357,  601,   507,   169;
  7: 1, 20, 160,  664, 1550,  2048,  1428,   408;
  8: 1, 23, 218, 1112, 3346,  6106,  6632,  3940,   985;
  9: 1, 26, 285, 1728, 6394, 15012, 22442, 20680, 10701, 2378;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (3*n^2-6*n+4+(20*n^2-51*n+36)*x+(58*n^2-169*n+120)*x^2+(92*n^2-285*n+200)*x^3+(83*n^2-261*n+180)*x^4+(40*n^2-124*n+84)*x^5+(8*n^2-24*n+16)*x^6)*P(n-1,x) - (n-1+(14*n-25)*x+(70*n-142)*x^2+(176*n-370)*x^3+(249*n-525)*x^4+(202*n-421)*x^5+(88*n-180)*x^6+(16*n-32)*x^7)*d/dx P(n-1,x) + (x+13*x^2+58*x^3+130*x^4+165*x^5+121*x^6+48*x^7+8*x^8)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 6*n + 4 + (20*n^2 - 51*n + 36)*x + (58*n^2 - 169*n + 120)*x^2 + (92*n^2 - 285*n + 200)*x^3 + (83*n^2 - 261*n + 180)*x^4 + (40*n^2 - 124*n + 84)*x^5 + (8*n^2 - 24*n + 16)*x^6)*P[n - 1, x] - (n - 1 + (14*n - 25)*x + (70*n - 142)*x^2 + (176*n - 370)*x^3 + (249*n - 525)*x^4 + (202*n - 421)*x^5 + (88*n - 180)*x^6 + (16*n - 32)*x^7)*D[P[n - 1, x], x] + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^2 - 6*n + 4 + (20*n^2 - 51*n + 36)*x + (58*n^2 - 169*n + 120)*x^2 + (92*n^2 - 285*n + 200)*x^3 + (83*n^2 - 261*n + 180)*x^4 + (40*n^2 - 124*n + 84)*x^5 + (8*n^2 - 24*n + 16)*x^6)*v[i - 1] - (n - 1 + (14*n - 25)*x + (70*n - 142)*x^2 + (176*n - 370)*x^3 + (249*n - 525)*x^4 + (202*n - 421)*x^5 + (88*n - 180)*x^6 + (16*n - 32)*x^7)*deriv(v[i - 1], x) + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*x^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)

Generated with polytool c48629374597.

A209695: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  5,   5;
   4: 1,  8,  18,   12;
   5: 1, 11,  40,   58,   29;
   6: 1, 14,  71,  164,  175,    70;
   7: 1, 17, 111,  357,  601,   507,   169;
   8: 1, 20, 160,  664, 1550,  2048,  1428,   408;
   9: 1, 23, 218, 1112, 3346,  6106,  6632,  3940,   985;
  10: 1, 26, 285, 1728, 6394, 15012, 22442, 20680, 10701, 2378;
%F Formula
The row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (3*n^2-12*n+13+(20*n^2-91*n+107)*x+(58*n^2-285*n+347)*x^2+(92*n^2-469*n+577)*x^3+(83*n^2-427*n+524)*x^4+(40*n^2-204*n+248)*x^5+(8*n^2-40*n+48)*x^6)*P(n-1,x) - (n-2+(14*n-39)*x+(70*n-212)*x^2+(176*n-546)*x^3+(249*n-774)*x^4+(202*n-623)*x^5+(88*n-268)*x^6+(16*n-48)*x^7)*d/dx P(n-1,x) + (x+13*x^2+58*x^3+130*x^4+165*x^5+121*x^6+48*x^7+8*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 12*n + 13 + (20*n^2 - 91*n + 107)*x + (58*n^2 - 285*n + 347)*x^2 + (92*n^2 - 469*n + 577)*x^3 + (83*n^2 - 427*n + 524)*x^4 + (40*n^2 - 204*n + 248)*x^5 + (8*n^2 - 40*n + 48)*x^6)*P[n - 1, x] - (n - 2 + (14*n - 39)*x + (70*n - 212)*x^2 + (176*n - 546)*x^3 + (249*n - 774)*x^4 + (202*n - 623)*x^5 + (88*n - 268)*x^6 + (16*n - 48)*x^7)*D[P[n - 1, x], x] + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*D[P[n - 1, x], {x, 2}])/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n^2 - 12*n + 13 + (20*n^2 - 91*n + 107)*x + (58*n^2 - 285*n + 347)*x^2 + (92*n^2 - 469*n + 577)*x^3 + (83*n^2 - 427*n + 524)*x^4 + (40*n^2 - 204*n + 248)*x^5 + (8*n^2 - 40*n + 48)*x^6)*v[i - 1] - (n - 2 + (14*n - 39)*x + (70*n - 212)*x^2 + (176*n - 546)*x^3 + (249*n - 774)*x^4 + (202*n - 623)*x^5 + (88*n - 268)*x^6 + (16*n - 48)*x^7)*deriv(v[i - 1], x) + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)); 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 >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-019f8e2975c7c2ec
  • 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-019f8e2975c7c2ec
  • 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-019f8e2975c7c2ec

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

Same fingerprint (3)

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