basis-map coefficients of A297191, shifted → (θ+1)² → basis A013611

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Triangle begins:
  0:   1;
  1:   9,    72,    144;
  2:  25,   400,   2400,     6400,      6400;
  3:  49,  1176,  11760,    62720,    188160,     301056,      200704;
  4:  81,  2592,  36288,   290304,   1451520,    4644864,     9289728,     10616832,       5308416;
  5: 121,  4840,  87120,   929280,   6504960,   31223808,   104079360,    237895680,     356843520,     317194240,      126877696;
  6: 169,  8112, 178464,  2379520,  21415680,  137060352,   639614976,   2192965632,    5482414080,    9746513920,    11695816704,     8506048512,      2835349504;
  7: 225, 12600, 327600,  5241600,  57657600,  461260800,  2767564800,  12651724800,   44281036800,  118082764800,   236165529600,   343513497600,    343513497600,    211392921600,     60397977600;
  8: 289, 18496, 554880, 10357760, 134650880, 1292648448,  9479421952,  54168125440,  243756564480,  866690007040,  2426732019712,  5294688043008,   8824480071680,  10860898549760,   9309341614080,   4964982194176,   1241245548544;
  9: 361, 25992, 883728, 18852864, 282792960, 3167281152, 27449769984, 188226994176, 1035248467968, 4601104302080, 16563975487488, 48186110509056, 112434257854464, 207570937577472, 296529910824960, 316298571546624, 237223928659968, 111634789957632, 24807731101696;

The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = (4*n^2+4*n+1+(32*n^2+32*n+8)*x+(64*n^2+64*n+16)*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 + (32*n^2 + 32*n + 8)*x + (64*n^2 + 64*n + 16)*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 + (32*n^2 + 32*n + 8)*x + (64*n^2 + 64*n + 16)*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 (−∞, 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_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)

Related ?

Same fingerprint ?

• basis-map coefficients of A297191, shifted → θ+1 → basis A013611
• basis-map coefficients of A297191, shifted → basis A013611 → reverse
• basis-map coefficients of A297191, shifted → basis A013611

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.