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

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Submission text ?

Triangle begins:
  0:  1;
  1:  3,   24,    48;
  2:  5,   80,   480,   1280,     1280;
  3:  7,  168,  1680,   8960,    26880,     43008,      28672;
  4:  9,  288,  4032,  32256,   161280,    516096,    1032192,    1179648,      589824;
  5: 11,  440,  7920,  84480,   591360,   2838528,    9461760,   21626880,    32440320,     28835840,     11534336;
  6: 13,  624, 13728, 183040,  1647360,  10543104,   49201152,  168689664,   421724160,    749731840,    899678208,     654311424,     218103808;
  7: 15,  840, 21840, 349440,  3843840,  30750720,  184504320,  843448320,  2952069120,   7872184320,  15744368640,   22900899840,   22900899840,    14092861440,     4026531840;
  8: 17, 1088, 32640, 609280,  7920640,  76038144,  557613056, 3186360320, 14338621440,  50981765120, 142748942336,  311452237824,  519087063040,   638876385280,   547608330240,   292057776128,    73014444032;
  9: 19, 1368, 46512, 992256, 14883840, 166699008, 1444724736, 9906683904, 54486761472, 242163384320, 871788183552, 2536111079424, 5917592518656, 10924786188288, 15606837411840, 16647293239296, 12485469929472, 5875515260928, 1305670057984;

The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(16*n+8)*x+(32*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[((2*n + 1 + (16*n + 8)*x + (32*n + 16)*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 + (16*n + 8)*x + (32*n + 16)*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 → (θ+1)² → 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.