A038226 → basis A013610 → D+1

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Triangle begins:
  0:           649,            1680,              576;
  1:         10043,           46728,            60480,            13824;
  2:        142417,          964128,          2242944,          1935360,            331776;
  3:       1917971,        17090040,         57847680,         89717760,          58060800,           7962624;
  4:      24962905,       276187824,       1230482880,       2776688640,        3229839360,        1672151040,          191102976;
  5:     317109419,      4193768040,      23199777216,      68907041280,      116620922880,      108522602496,        46820229120,         4586471424;
  6:    3955895713,     60885008448,     402601731840,    1484785741824,     3307537981440,     4478243438592,      3472723279872,      1284211998720,       110075314176;
  7:   48659465987,    854473474008,    6575580912384,   28987324692480,    80178430058496,   142885640798208,    161216763789312,    107158318350336,     34673723965440,     2641807540224;
  8:  591844870441,  11678271836880,  102536816880960,  526046472990720,  1739239481548800,  3848564642807808,   5715425631928320,   5527431901347840,   3214749550510080,   924632639078400,    63403380965376;
  9: 7132791765275, 156247045796424, 1541531882468160, 9023239885524480, 34719067217387520, 91831844625776640, 169336844283543552, 215553195261296640, 182405252744478720, 94299320148295680, 24410301671669760, 1521681143169024;

The row polynomials satisfy (24*n+35+24*x)*P(n,x) = (264*n+649+(576*n+1680)*x+576*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 649 + 1680*x + 576*x^2.

P[0, x_] := 649 + 1680*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((264*n + 649 + (576*n + 1680)*x + 576*x^2)*P[n - 1, x])/(24*n + 35 + 24*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 649 + 1680*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((264*n + 649 + (576*n + 1680)*x + 576*x^2)*v[i - 1])/(24*n + 35 + 24*x)); 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013610 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.?
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                • 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
            • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.