A038333 → basis A013611 → D

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Submission text ?

Triangle begins:
  0:             1064,              1568;
  1:            30324,             89376,              65856;
  2:           768208,           3396288,            5005056,            2458624;
  3:         18244940,         107549120,          237740160,          233569280,            86051840;
  4:        415984632,        3065149920,         9034126080,        13313448960,          9809909760,          2891341824;
  5:       9220992676,       81532987872,       300384692160,       590229570560,        652358999040,        384548462592,         94450499584;
  6:     200227269536,     2065502359424,      9131694641664,     22428723681280,      33052855951360,      29225683156992,      14356475936768,       3022415986688;
  7:    4279857886332,    50457271923072,    260253297287424,    767062349899776,    1413009591920640,    1665863939948544,    1227478692593664,     516833133723648,      95206103580672;
  8:   90352555378120,  1198360208172960,   7064018069230080,  24290307746826240,   53694364492984320,   79128537147555840,   77740317197598720,   49099147703746560,   18089159680327680,   2961967666954240;
  9: 1888368407402708, 27828587056460960, 184547472058635840, 725239188440954880, 1870353696505620480, 3307572852767834112, 4061931573574533120, 3420573956694343680, 1890317186594242560, 619051242393436160, 91228604142190592;

The row polynomials satisfy (n+1)*P(n,x) = (19*n+38+(28*n+56)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1064 + 1568*x.

P[0, x_] := 1064 + 1568*x;
P[n_, x_] := P[n, x] = Expand[Together[((19*n + 38 + (28*n + 56)*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] = 1064 + 1568*x; for(n = 1, N, my(i = n + 1); v[i] = ((19*n + 38 + (28*n + 56)*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)*12^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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 ?

• A038333 → basis A013611
• A038333 → basis A013611 → 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.