A038328 → basis A013609 → D

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Submission text ?

Triangle begins:
  0:            112,             32;
  1:           2352,           1344,            192;
  2:          43904,          37632,          10752,           1024;
  3:         768320,         878080,         376320,          71680,           5120;
  4:       12907776,       18439680,       10536960,        3010560,         430080,         24576;
  5:      210827008,      361417728,      258155520,       98344960,       21073920,       2408448,        114688;
  6:     3373232128,     6746464256,     5782683648,     2753658880,      786759680,     134873088,      12845056,       524288;
  7:    53128406016,   121436356608,   121436356608,    69392203776,    24782929920,    5664669696,     809238528,     66060288,     2359296;
  8:   826441871360,  2125136240640,  2428727132160,  1619151421440,   693922037760,  198263439360,   37764464640,   4624220160,   330301440,   10485760;
  9: 12727204818944, 36363442339840, 46752997294080, 35621331271680, 17810665635840, 6106513932288, 1453931888640, 237376634880, 25433210880, 1614807040, 46137344;

The row polynomials satisfy (n+1)*P(n,x) = (14*n+28+(4*n+8)*x)*P(n-1,x) for n >= 1, with P(0,x) = 112 + 32*x.

P[0, x_] := 112 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((14*n + 28 + (4*n + 8)*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] = 112 + 32*x; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 28 + (4*n + 8)*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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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 ?

• A038291 → basis A013611 → basis A007318
• A038334 → x ↦ x+1 → x ↦ x+1
• A038329 → basis A013609 → basis A007318
• A038255 → basis A013609 → D
• A013622 → basis A013611 → x ↦ x/(x+1)
• A038255 → basis A013609 → reverse
• A013612 → basis A013610 → x ↦ x/(x+1)
• A038328 → basis A013609
• A038255 → basis A013609
• A038328 → basis A013609 → 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.