basis-map coefficients of A026729, shifted → θ+1 → basis A013611

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Triangle begins:
   1:   1;
   2:  -2,   -8;
   3:   3,   24,    48;
   4:  -4,  -48,  -192,   -256;
   5:   5,   80,   480,   1280,    1280;
   6:  -6, -120,  -960,  -3840,   -7680,    -6144;
   7:   7,  168,  1680,   8960,   26880,    43008,    28672;
   8:  -8, -224, -2688, -17920,  -71680,  -172032,  -229376,  -131072;
   9:   9,  288,  4032,  32256,  161280,   516096,  1032192,  1179648,   589824;
  10: -10, -360, -5760, -53760, -322560, -1290240, -3440640, -5898240, -5898240, -2621440;

The row polynomials satisfy (n-1)*P(n,x) = -(n+4*n*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 4*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

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

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= …?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= …?
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

• A131431 → D → basis A013611
• A131436 → basis A013611
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013611
• A027466 → basis A013609 → D
• basis-map coefficients of A021012 → basis A013611
• basis-map coefficients of A112227 → basis A013611 → D
• basis-map coefficients of A026729 → D → basis A013611
• A038231 → D
• basis-map coefficients of A021012 → θ+1 → basis A013611
• A136572 → basis A013611 → D
• A038330 → basis A007318
• A132440 → θ+1 → basis A013611
• A130123 → D → basis A013611
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• A013611 → D
• A038303 → basis A007318 → basis A013610
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• basis-map coefficients of A026729, shifted → D → basis A013611
• image under θ+1 → θ+1 → D → basis A013611
• A130123 → θ+1 → basis A013611
• basis-map coefficients of A112227 → basis A013611
• A127648 → D → basis A013611
• A132440 → basis A013611
• A027467 → basis A013611 → D
• A038330 → basis A007318 → D
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013611
• A131431 → θ+1 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013611
• A136572 → basis A013611
• A038329 → D
• basis-map coefficients of A026729 → (θ+1)² → basis A013611
• A136572 → (θ+1)² → basis A013611
• A134309 → basis A013611
• A136572 → basis A013611 → reverse
• A130123 → (θ+1)² → basis A013611
• A131431 → basis A013611
• A129186 → basis A013611
• A027466 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• A120910 → basis A007318 → basis A007318
• image under θ+1 → basis A013611
• A130123 → basis A013611 → D
• A130123 → basis A013611
• A136521 → basis A013611
• basis-map coefficients of A026729 → basis A013611
• A027467 → basis A013611 → reverse
• A038329 → reverse
• image under θ+1 → (θ+1)² → basis A013611
• A136572 → D → basis A013611
• A038231 → D → D
• basis-map coefficients of A026729 → basis A013611 → reverse
• A013611 → D → D
• A136572 → θ+1 → basis A013611
• basis-map coefficients of A112227 → θ+1 → basis A013611
• basis-map coefficients of A030528, shifted → θ+1 → basis A013611
• basis-map coefficients of A112227 → basis A013611 → reverse
• A027467 → basis A013611
• A013610 → basis A013611 → x ↦ x/(x+1)
• A038330 → basis A007318 → reverse
• basis-map coefficients of A026729 → basis A013611 → D
• basis-map coefficients of A112227 → (θ+1)² → basis A013611
• basis-map coefficients of A030528, shifted → D → basis A013611
• basis-map coefficients of A026729, shifted → basis A013611

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.