basis-map coefficients of A297191, shifted → θ+1 → basis A013610

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Submission text ?

Triangle begins:
  0:  1;
  1:  3,   18,    27;
  2:  5,   60,   270,    540,     405;
  3:  7,  126,   945,   3780,    8505,    10206,      5103;
  4:  9,  216,  2268,  13608,   51030,   122472,    183708,     157464,      59049;
  5: 11,  330,  4455,  35640,  187110,   673596,   1683990,    2886840,    3247695,     2165130,      649539;
  6: 13,  468,  7722,  77220,  521235,  2501928,   8756748,   22517352,   42220035,    56293380,    50664042,     27634932,      6908733;
  7: 15,  630, 12285, 147420, 1216215,  7297290,  32837805,  112586760,  295540245,   591080490,   886620735,    967222620,    725416965,    334807830,     71744535;
  8: 17,  816, 18360, 257040, 2506140, 18044208,  99243144,  425327760, 1435481190,  3827949840,  8038694664,  13154227632,  16442784540,  15177954960,   9757256760,   3902902704,    731794257;
  9: 19, 1026, 26163, 418608, 4709340, 39558456, 257129964, 1322382672, 5454828522, 18182761740, 49093456698, 107112996432, 187447743756, 259543029816, 278081817660, 222465454128, 125136817947, 44165935746, 7360989291;

The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(12*n+6)*x+(18*n+9)*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 + (12*n + 6)*x + (18*n + 9)*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 + (12*n + 6)*x + (18*n + 9)*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 (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from 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)

Related ?

Same fingerprint ?

• basis-map coefficients of A297191, shifted → basis A013610 → reverse
• A139548 → x ↦ x+1 → basis A007318
• A034870 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A297191, shifted → basis A013610
• basis-map coefficients of A297191, shifted → (θ+1)² → 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.