A038305 → x ↦ x+1 → D

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Submission text ?

Triangle begins:
  0:            78,             18;
  1:          1521,            702,             81;
  2:         26364,          18252,           4212,           324;
  3:        428415,         395460,         136890,         21060,          1215;
  4:       6683274,        7711470,        3559140,        821340,         94770,         4374;
  5:     101362989,      140348754,       80970435,      24913980,       4312035,       398034,        15309;
  6:    1505964408,     2432711736,     1684185048,     647763480,     149483880,     20697768,      1592136,       52488;
  7:   22024729467,    40661039016,    32841608436,   15157665432,    4372403490,    807212952,     93139956,     6141096,     177147;
  8:  318134981190,   660741884010,   609915585240,  328416084360,  113682490740,  26234420940,   4036064760,   399171240,   23029110,   590490;
  9: 4549330231017, 10498454379270, 10902241086165, 6709071437640, 2709432695970, 750304438884, 144289315170, 19027162440, 1646581365, 84440070, 1948617;

The row polynomials satisfy (n+1)*P(n,x) = (13*n+26+(3*n+6)*x)*P(n-1,x) for n >= 1, with P(0,x) = 78 + 18*x.

P[0, x_] := 78 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((13*n + 26 + (3*n + 6)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 78 + 18*x; for(n = 1, N, my(i = n + 1); v[i] = ((13*n + 26 + (3*n + 6)*x)*v[i - 1])/(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 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)*10^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A038305 → x ↦ x+1
• A013612 → basis A013611 → x ↦ x/(x+1)
• A038305 → x ↦ x+1 → reverse
• A038327 → basis A013610 → D
• A038327 → basis A013610
• A038327 → basis A013610 → reverse

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.