basis-map coefficients of A297191, shifted → (θ+1)² → basis A007318

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Triangle begins:
  0:   1;
  1:   9,   18,     9;
  2:  25,  100,   150,    100,      25;
  3:  49,  294,   735,    980,     735,     294,      49;
  4:  81,  648,  2268,   4536,    5670,    4536,    2268,      648,       81;
  5: 121, 1210,  5445,  14520,   25410,   30492,   25410,    14520,     5445,     1210,      121;
  6: 169, 2028, 11154,  37180,   83655,  133848,  156156,   133848,    83655,    37180,    11154,     2028,     169;
  7: 225, 3150, 20475,  81900,  225225,  450450,  675675,   772200,   675675,   450450,   225225,    81900,   20475,    3150,     225;
  8: 289, 4624, 34680, 161840,  525980, 1262352, 2314312,  3306160,  3719430,  3306160,  2314312,  1262352,  525980,  161840,   34680,   4624,   289;
  9: 361, 6498, 55233, 294576, 1104660, 3093048, 6701604, 11488464, 15796638, 17551820, 15796638, 11488464, 6701604, 3093048, 1104660, 294576, 55233, 6498, 361;

The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = (4*n^2+4*n+1+(8*n^2+8*n+2)*x+(4*n^2+4*n+1)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 4*n + 1 + (8*n^2 + 8*n + 2)*x + (4*n^2 + 4*n + 1)*x^2)*P[n - 1, x])/(4*n^2 - 4*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 4*n + 1 + (8*n^2 + 8*n + 2)*x + (4*n^2 + 4*n + 1)*x^2)*v[i - 1])/(4*n^2 - 4*n + 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Del…?
        • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Del…?
            • ↑ Laguerre’s theorem on multiplier sequences (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of…?
        • ↑ Laguerre’s theorem on multiplier sequences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of…?
    • ↑ Laguerre’s theorem on multiplier sequences (above)
    • ↑ Laguerre’s theorem on multiplier sequences (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A139548 → x ↦ x+1 → basis A013610
• A034870: Even-numbered rows of Pascal's triangle.
• A034870 → basis A013609
• A034870 → basis A013609 → basis A013609
• A139548 → basis A013609 → basis A013609
• basis-map coefficients of A297191, shifted → θ+1 → x ↦ x+1
• 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.
• A034870 → D → D
• A139548 → basis A013609
• A034870 → x ↦ x+1 → basis A013610

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.