A131436 → basis A008277

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Submission text ?

Triangle begins:
  0:    1;
  1:    3,      3;
  2:    7,     21,       7;
  3:   15,    105,      90,       15;
  4:   31,    465,     775,      310,       31;
  5:   63,   1953,    5670,     4095,      945,       63;
  6:  127,   8001,   38227,    44450,    17780,     2667,     127;
  7:  255,  32385,  246330,   433755,   267750,    67830,    7140,    255;
  8:  511, 130305, 1545775,  3970470,  3551961,  1352106,  236082,  18396,   511;
  9: 1023, 522753, 9544590, 34889415, 43503075, 23352021, 6015240, 767250, 46035, 1023;

The row polynomials satisfy P(n,x) = -(2+6*x+2*x^2)*P(n-2,x) - (6*x+4*x^2)*d/dx P(n-2,x) - 2*x^2*d^2/dx^2 P(n-2,x) + (3+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.

P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(2 + 6*x + 2*x^2)*P[n - 2, x] - (6*x + 4*x^2)*D[P[n - 2, x], x] - 2*x^2*D[P[n - 2, x], {x, 2}] + (3 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 6*x + 2*x^2)*v[i - 2] - (6*x + 4*x^2)*deriv(v[i - 2], x) - 2*x^2*deriv(deriv(v[i - 2], x), x) + (3 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); 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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from Triangle read by rows, (n-1) zeros followed by 2^n - 1.?
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows, (n-1) zeros followed by 2^n - 1.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A112227 → (θ+1)² → basis A008277
• A130123 → (θ+1)² → basis A008277
• A048993 → D+1 → reverse
• basis-map coefficients of A026729, shifted → basis A008277
• basis-map coefficients of A030528, shifted → D → basis A008277
• A136521 → basis A008277
• basis-map coefficients of A026729, shifted → θ+1 → basis A008277
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A008277
• basis-map coefficients of A026729 → (θ+1)² → basis A008277
• A136572 → (θ+1)² → basis A008277
• A048993 → D+1
• A127648 → D → basis A008277
• basis-map coefficients of A030528, shifted → θ+1 → basis A008277
• A048993 → reverse
• image under θ+1 → (θ+1)² → basis A008277
• A130123 → θ+1 → basis A008277
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A008277
• basis-map coefficients of A021012 → basis A008277
• A106800: Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
• A129186 → basis A008277
• basis-map coefficients of A021012 → basis A008277 → reverse
• A134309 → basis A008277
• A008277: Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
• A131431 → D → basis A008277
• A136572 → basis A008277
• A048993: Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
• A008278: Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
• basis-map coefficients of A112227 → θ+1 → basis A008277
• A130123 → D → basis A008277
• A130123 → basis A008277
• A136572 → θ+1 → basis A008277
• basis-map coefficients of A021012 → θ+1 → basis A008277
• A131431 → θ+1 → basis A008277
• A132440 → θ+1 → basis A008277
• A132440 → basis A008277
• basis-map coefficients of A030528, shifted → basis A008277
• A131431 → basis A008277
• A136572 → D → basis A008277
• basis-map coefficients of A026729, shifted → D → basis A008277
• basis-map coefficients of A026729 → D → basis A008277
• basis-map coefficients of A112227 → basis A008277
• image under θ+1 → basis A008277
• A130123 → basis A008277 → reverse
• basis-map coefficients of A112227 → basis A008277 → reverse
• image under θ+1 → θ+1 → D → basis A008277

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
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741