basis-map coefficients of A297191, shifted → θ+1 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:  1;
  1:  3,   6,    3;
  2:  5,  20,   30,    20,     5;
  3:  7,  42,  105,   140,   105,     42,      7;
  4:  9,  72,  252,   504,   630,    504,    252,     72,      9;
  5: 11, 110,  495,  1320,  2310,   2772,   2310,   1320,    495,    110,     11;
  6: 13, 156,  858,  2860,  6435,  10296,  12012,  10296,   6435,   2860,    858,    156,     13;
  7: 15, 210, 1365,  5460, 15015,  30030,  45045,  51480,  45045,  30030,  15015,   5460,   1365,    210,    15;
  8: 17, 272, 2040,  9520, 30940,  74256, 136136, 194480, 218790, 194480, 136136,  74256,  30940,   9520,  2040,   272,   17;
  9: 19, 342, 2907, 15504, 58140, 162792, 352716, 604656, 831402, 923780, 831402, 604656, 352716, 162792, 58140, 15504, 2907, 342, 19;

The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(4*n+2)*x+(2*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[((2*n + 1 + (4*n + 2)*x + (2*n + 1)*x^2)*P[n - 1, x])/(2*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] = ((2*n + 1 + (4*n + 2)*x + (2*n + 1)*x^2)*v[i - 1])/(2*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 Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (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 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 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 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 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) (above)

Related ?

Same fingerprint ?

• A139548 → x ↦ x+1 → basis A013610
• A139548 → basis A013609 → basis A013609
• basis-map coefficients of A297191, shifted → (θ+1)² → basis A007318
• A034870 → D → D
• A139548 → basis A013609
• A034870 → basis A013609
• A034870 → x ↦ x+1 → basis A013610
• 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.
• 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.