A131436 → basis A013610

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Submission text ?

Triangle begins:
  0:    1;
  1:    3,     9;
  2:    7,    42,     63;
  3:   15,   135,    405,     405;
  4:   31,   372,   1674,    3348,     2511;
  5:   63,   945,   5670,   17010,    25515,    15309;
  6:  127,  2286,  17145,   68580,   154305,   185166,    92583;
  7:  255,  5355,  48195,  240975,   722925,  1301265,  1301265,   557685;
  8:  511, 12264, 128772,  772632,  2897370,  6953688, 10430532,  8940456,  3352671;
  9: 1023, 27621, 331452, 2320164, 10440738, 31322214, 62644428, 80542836, 60407127, 20135709;

The row polynomials satisfy P(n,x) = -(2+12*x+18*x^2)*P(n-2,x) + (3+9*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 9*x.

P[0, x_] := 1;
P[1, x_] := 3 + 9*x;
P[n_, x_] := P[n, x] = Expand[-(2 + 12*x + 18*x^2)*P[n - 2, x] + (3 + 9*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 12*x + 18*x^2)*v[i - 2] + (3 + 9*x)*v[i - 1]); 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? 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 ?

• A027465: Cube of lower triangular normalized binomial matrix.
• A013610 → D → D
• A131431 → D → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → D
• basis-map coefficients of A026729, shifted → D → basis A013610
• A028326 → basis A007318 → basis A007318
• basis-map coefficients of A112227 → basis A013610 → reverse
• A013609 → basis A013610 → x ↦ x/(x+1)
• A130123 → D → basis A013610
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013610
• basis-map coefficients of A026729 → basis A013610
• A356546 → basis A013611 → x ↦ x/(x+1)
• A127648 → basis A013610
• image under θ+1 → (θ+1)² → basis A013610
• A136572 → basis A013610 → basis A013611
• A178820 → basis A007318 → basis A007318
• A120910 → basis A007318
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610 → reverse
• basis-map coefficients of A112227 → basis A013610
• A130123 → basis A013610 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013610 → D
• basis-map coefficients of A112227 → (θ+1)² → basis A013610
• basis-map coefficients of A112227 → θ+1 → basis A013610
• A156992 → basis A013611 → x ↦ x/(x+1)
• A136572 → (θ+1)² → basis A013610
• A013610 → basis A013611 → D
• A178822 → basis A013611 → x ↦ x/(x+1)
• A196347 → x ↦ x+1 → basis A007318
• A038279 → basis A013610
• A130123 → θ+1 → basis A013610
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013610
• A120909 → basis A013610 → x ↦ x/(x+1)
• A136572 → θ+1 → basis A013610
• A131431 → θ+1 → basis A013610
• A038338 → x ↦ x+1 → basis A007318
• basis-map coefficients of A026729, shifted → θ+1 → basis A013610
• basis-map coefficients of A021012 → θ+1 → basis A013610
• A136572 → D → basis A013610
• A135065 → x ↦ x+1 → basis A007318
• A038329 → x ↦ x+1 → basis A013609
• A136572 → basis A013610
• A085881 → basis A007318 → basis A007318
• A038243 → basis A013609 → D
• basis-map coefficients of A026729 → D → basis A013610
• basis-map coefficients of A030528, shifted → θ+1 → basis A013610
• basis-map coefficients of A112227 → basis A013610 → basis A013611
• A038279 → basis A013610 → D
• basis-map coefficients of A026729, shifted → basis A013610 → reverse
• A027465 → D
• A178822 → x ↦ x+1 → x ↦ x+1
• basis-map coefficients of A026729, shifted → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• A013610 → basis A013611
• A085881 → basis A013611 → x ↦ x/(x+1)
• A120910 → basis A007318 → reverse
• A127648 → D → basis A013610
• A013610 → basis A013611 → reverse
• A178821 → basis A013611 → x ↦ x/(x+1)
• image under θ+1 → basis A013610
• basis-map coefficients of A026729 → basis A013610 → D
• A027465 → D → D
• A134309 → basis A013610
• A085880 → basis A013611 → x ↦ x/(x+1)
• A178820 → basis A013611 → x ↦ x/(x+1)
• A120909 → x ↦ x/(x+1)
• A038330 → basis A013609 → x ↦ x+1
• A129186 → basis A013610
• A132440 → θ+1 → basis A013610
• A356546 → x ↦ x+1 → x ↦ x+1
• A130123 → (θ+1)² → basis A013610
• A085880 → x ↦ x+1 → x ↦ x+1
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
• A147716 → basis A013610 → basis A013609
• A094305 → x ↦ x+1 → x ↦ x+1
• basis-map coefficients of A026729 → (θ+1)² → basis A013610
• A013610 → D
• A038243 → basis A013609
• basis-map coefficients of A026729 → basis A013610 → basis A013611
• A130123 → basis A013610
• basis-map coefficients of A026729 → basis A013610 → reverse
• A038330 → reverse
• basis-map coefficients of A021012 → basis A013610
• A038325 → basis A007318 → basis A007318
• A156992 → basis A007318 → x ↦ x+1
• A136572 → basis A013610 → D
• A038330 → D
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013611
• basis-map coefficients of A112227 → basis A013610 → D
• A120910 → D → x ↦ x+1
• basis-map coefficients of A030528, shifted → basis A013610
• basis-map coefficients of A030528, shifted → D → basis A013610
• image under θ+1 → θ+1 → D → basis A013610
• A038279 → basis A013610 → reverse
• A130123 → basis A013610 → D
• A178821 → basis A007318 → x ↦ x+1
• A131431 → basis A013610
• A028326 → basis A013611 → x ↦ x/(x+1)
• A038221 → basis A013611 → x ↦ x/(x+1)
• A136521 → basis A013610
• A196347 → basis A013611 → x ↦ x/(x+1)
• A013610 → D → basis A013611

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