basis-map coefficients of A026729 → basis A007318 → θ+1

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Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  4,   3;
   4: 1,  6,   9,   4;
   5: 1,  8,  18,  16,   5;
   6: 1, 10,  30,  40,  25,   6;
   7: 1, 12,  45,  80,  75,  36,   7;
   8: 1, 14,  63, 140, 175, 126,  49,   8;
   9: 1, 16,  84, 224, 350, 336, 196,  64,  9;
  10: 1, 18, 108, 336, 630, 756, 588, 288, 81, 10;

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

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (n + 1)*x + n*x^2)*P[n - 1, x])/(1 + (n - 1)*x)]];
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] = ((1 + (n + 1)*x + n*x^2)*v[i - 1])/(1 + (n - 1)*x)); 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 Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
    • 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…?
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • 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…?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (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…?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
            • Parent-row provers of polytool polytool ? step › direct check
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
            • Parent-row provers of polytool polytool ? step › direct check
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A038221 → D+1
• A135838 → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1
• A028326 → D → D+1
• A028326 → D+1
• A178822 → basis A013609 → D+1
• A038208 → D → D+1
• A085881 → D+1
• basis-map coefficients of A113214, shifted → θ+1
• A027465 → basis A013611 → θ+1
• A038325 → D+1
• A094305 → θ+1
• A120910 → basis A013610 → θ+1
• basis-map coefficients of A113214, shifted → D → θ+1
• A028326 → D → θ+1
• A124860 → D+1
• basis-map coefficients of A026729 → basis A007318 → D+1
• basis-map coefficients of A113214, shifted → D+1
• A094305 → D+1
• A038325 → basis A013609 → θ+1
• A085880 → D → D+1
• A085880 → basis A013609 → θ+1
• basis-map coefficients of A113214, shifted → basis A013609 → D+1
• A038325 → basis A013609 → D+1
• A178821 → basis A013609 → θ+1
• basis-map coefficients of A127647 → θ+1
• A038338 → D+1
• A038221 → basis A013609 → D+1
• A124860 → θ+1
• A038208 → θ+1
• basis-map coefficients of A026729, shifted → basis A007318 → D+1
• A131431 → x ↦ x+1 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A093375 → D
• A085881 → θ+1
• A196347 → basis A013609 → θ+1
• A356546 → basis A013609 → D+1
• A196347 → θ+1
• A085880 → D → θ+1
• A178821 → D+1
• basis-map coefficients of A127647 → D+1
• A003506 → D → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → D+1
• A178821 → θ+1
• A085880 → θ+1
• A120910 → basis A013610 → D+1
• A038338 → basis A013609 → D+1
• A156992 → D+1
• A028326 → basis A013609 → D+1
• A085880 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A038208 → D+1
• A038330 → basis A013611 → D+1
• A156992 → basis A013609 → θ+1
• A085880 → basis A013609 → D+1
• A085881 → basis A013609 → θ+1
• A216973 → θ+1
• A178820 → basis A013609 → D+1
• A156992 → θ+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
• A134346 → D+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A376827 → θ+1
• A344565 → D+1
• A178820 → D+1
• A085881 → basis A013609 → D+1
• A178822 → basis A013609 → θ+1
• A038208 → D → θ+1
• A135065 → D+1
• A135838 → θ+1
• A134346 → θ+1
• A028326 → θ+1
• A038221 → θ+1
• A376827 → D+1
• A178821 → basis A013609 → D+1
• A178820 → basis A013609 → θ+1
• A028326 → basis A013609 → θ+1
• A196347 → basis A013609 → D+1
• A093375 → D → D
• basis-map coefficients of A113214, shifted → D → D+1
• A178822 → D+1
• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
• A027465 → basis A013611 → D+1
• A356546 → θ+1
• A356546 → D+1
• A178820 → θ+1
• A344565 → θ+1
• A178822 → θ+1
• A156992 → basis A013609 → D+1
• A038338 → basis A013609 → θ+1
• A038221 → basis A013609 → θ+1
• A038330 → basis A013611 → θ+1
• A135065 → θ+1
• A131431 → x ↦ x+1 → θ+1
• A038338 → θ+1
• A131431 → D+1 → basis A007318
• A038325 → θ+1
• A196347 → D+1
• A356546 → basis A013609 → θ+1

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.