A034870 → D → D

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Submission text ?

Triangle begins:
  0:   0;
  1:   2;
  2:  12,   24,    12;
  3:  30,  120,   180,    120,     30;
  4:  56,  336,   840,   1120,    840,     336,      56;
  5:  90,  720,  2520,   5040,   6300,    5040,    2520,     720,      90;
  6: 132, 1320,  5940,  15840,  27720,   33264,   27720,   15840,    5940,    1320,     132;
  7: 182, 2184, 12012,  40040,  90090,  144144,  168168,  144144,   90090,   40040,   12012,    2184,    182;
  8: 240, 3360, 21840,  87360, 240240,  480480,  720720,  823680,  720720,  480480,  240240,   87360,  21840,   3360,   240;
  9: 306, 4896, 36720, 171360, 556920, 1336608, 2450448, 3500640, 3938220, 3500640, 2450448, 1336608, 556920, 171360, 36720, 4896, 306;

The row polynomials satisfy (2*n^2-5*n+3)*P(n,x) = (2*n^2-n+(4*n^2-2*n)*x+(2*n^2-n)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - n + (4*n^2 - 2*n)*x + (2*n^2 - n)*x^2)*P[n - 1, x])/(2*n^2 - 5*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 - n + (4*n^2 - 2*n)*x + (2*n^2 - n)*x^2)*v[i - 1])/(2*n^2 - 5*n + 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)

Properties

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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from derivative of Even-numbered rows of Pascal's triangle.?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Even-numbered rows of Pascal's triangle.?
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Even-numbered rows of Pascal's triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • Geometric basis map (elementary) polytool ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • 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
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • 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
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delanno…?
        • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

          • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delanno…?
        • ↑ Laguerre’s theorem on differential operators (above)

Related ?

Same fingerprint ?

• A139548 → x ↦ x+1 → basis A013610
• A034870 → basis A013609
• A034870 → basis A013609 → basis A013609
• A139548: Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.
• A139548 → basis A013609
• basis-map coefficients of A297191, shifted → θ+1 → x ↦ x+1
• basis-map coefficients of A297191, shifted → (θ+1)² → basis A007318
• A034870 → x ↦ x+1 → basis A013610
• A139548 → basis A013609 → basis A013609
• A034870: Even-numbered rows of Pascal's triangle.

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.