A038281 → x ↦ x+1 → D+1

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Submission text ?

Triangle begins:
  0:           187,            84,             9;
  1:          2420,          1683,           378,            27;
  2:         30613,         29040,         10098,          1512,            81;
  3:        380666,        459195,        217800,         50490,          5670,          243;
  4:       4670479,       6851988,       4132755,       1306800,        227205,        20412,          729;
  5:      56689952,      98080059,      71945874,      28929285,       6860700,       954261,        71442,        2187;
  6:     682050985,    1360558848,    1176960708,     575566992,     173575710,     32931360,      3817044,      244944,       6561;
  7:    8145637478,   18415376595,   18367544448,   10592646372,    3885077196,    937308834,    148191120,    14722884,     826686,     19683;
  8:   96675855331,  244369124340,  276230648925,  183675444480,   79444847790,  23310463176,   4686544170,   635104800,   55210815,   2755620,   59049;
  9: 1141246682444, 3190303225923, 4032090551610, 3038537138175, 1515322416960, 524335995414, 128207547468, 22093708230, 2619807300, 202439655, 9093546, 177147;

The row polynomials satisfy (3*n+14+3*x)*P(n,x) = (33*n+187+(9*n+84)*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 187 + 84*x + 9*x^2.

P[0, x_] := 187 + 84*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((33*n + 187 + (9*n + 84)*x + 9*x^2)*P[n - 1, x])/(3*n + 14 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 187 + 84*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((33*n + 187 + (9*n + 84)*x + 9*x^2)*v[i - 1])/(3*n + 14 + 3*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_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.?
    • Geometric basis map (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) [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) (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)
    • 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) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A038303 → basis A013610 → D+1

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.