A038284 → basis A013611 → D

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Submission text ?

Triangle begins:
  0:            672,             1152;
  1:          14112,            48384,             41472;
  2:         263424,          1354752,           2322432,           1327104;
  3:        4609920,         31610880,          81285120,          92897280,           39813120;
  4:       77446656,        663828480,        2275983360,        3901685760,         3344302080,         1146617856;
  5:     1264962048,      13011038208,       55761592320,      127455068160,       163870801920,       112368549888,        32105299968;
  6:    20239392768,     242872713216,     1249059667968,     3568741908480,      6117843271680,      6292638793728,      3595793596416,       880602513408;
  7:   318770436096,    4371708837888,    26230253027328,    89932296093696,    192712063057920,    264290829336576,    226534996574208,    110955916689408,     23776267862016;
  8:  4958651228160,   76504904663040,   524605060546560,  2098420242186240,   5395937765621760,   9250179026780160,  10571633173463040,   7766914168258560,   3328677500682240,   634033809653760;
  9: 76363228913664, 1309083924234240, 10098647415521280, 46165245328097280, 138495735984291840, 284905514024828928, 407007877178327040, 398701593970606080, 256308167552532480, 97641206686679040, 16738492574859264;

The row polynomials satisfy (n+1)*P(n,x) = (14*n+28+(24*n+48)*x)*P(n-1,x) for n >= 1, with P(0,x) = 672 + 1152*x.

P[0, x_] := 672 + 1152*x;
P[n_, x_] := P[n, x] = Expand[Together[((14*n + 28 + (24*n + 48)*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] = 672 + 1152*x; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 28 + (24*n + 48)*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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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
        • 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

Related ?

Same fingerprint ?

• A038335 → basis A013611
• A038271 → x ↦ x/(x+1) → reverse
• A038284 → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
• A038271 → x ↦ x/(x+1) → D
• A038271 → x ↦ x/(x+1)
• A038333 → D
• A038284 → basis A013611 → reverse
• A120910 → basis A013611 → basis A013610
• A038333 → 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.