A038329 → basis A013609 → D+1

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Submission text ?

Triangle begins:
  0:            405,             252,              36;
  1:           7425,            7290,            2268,             216;
  2:         131625,          178200,           87480,           18144,            1296;
  3:        2278125,         3948750,         2673000,          874800,          136080,            7776;
  4:       38728125,        82012500,        71077500,        32076000,         7873200,          979776,          46656;
  5:      649265625,      1626581250,      1722262500,       995085000,       336798000,        66134880,        6858432,         279936;
  6:    10764140625,     31164750000,     39037950000,     27556200000,     11941020000,      3233260800,      529079040,       47029248,       1679616;
  7:   176839453125,    581263593750,    841448250000,    702683100000,    372008700000,    128963016000,    29099347200,     4081466880,     317447424,     10077696;
  8:  2883251953125,  10610367187500,  17437907812500,  16828965000000,  10540246500000,   4464104400000,  1289630160000,   249422976000,   30611001600,   2116316160,    60466176;
  9: 46708681640625, 190294628906250, 350142117187500, 383633971875000, 277677922500000, 139131253800000, 49105148400000, 12159370080000, 2057739552000, 224480678400, 13967686656, 362797056;

The row polynomials satisfy (2*n+7+2*x)*P(n,x) = (30*n+135+(12*n+84)*x+12*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 405 + 252*x + 36*x^2.

P[0, x_] := 405 + 252*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((30*n + 135 + (12*n + 84)*x + 12*x^2)*P[n - 1, x])/(2*n + 7 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 405 + 252*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((30*n + 135 + (12*n + 84)*x + 12*x^2)*v[i - 1])/(2*n + 7 + 2*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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle of coefficients in expansion of (1+5x)^n.?
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • 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
            • 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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
            • Geometric basis map (elementary) polytool ? step ›

              • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(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
                • 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
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038291 → basis A013611 → D+1
• A038220 → basis A007318 → D+1
• A038334 → x ↦ x+1 → 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.