A038331 → basis A013611 → D

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Submission text ?

Triangle begins:
  0:             680,              800;
  1:           17340,            40800,             24000;
  2:          393040,          1387200,           1632000,            640000;
  3:         8352100,         39304000,          69360000,          54400000,           16000000;
  4:       170382840,       1002252000,        2358240000,        2774400000,         1632000000,          384000000;
  5:      3379259660,      23853597600,       70157640000,      110051200000,        97104000000,        45696000000,         8960000000;
  6:     65654187680,     540681545600,     1908287808000,     3741740800000,      4402048000000,      3107328000000,      1218560000000,       204800000000;
  7:   1255636339380,   11817753782400,    48661339104000,   114497268480000,    168378336000000,    158473728000000,     93219840000000,     31334400000000,     4608000000000;
  8:  23717575299400,  251127267876000,  1181775378240000,  3244089273600000,   5724863424000000,   6735133440000000,   5282457600000000,   2663424000000000,   783360000000000,   102400000000000;
  9: 443518658098780, 5217866565868000, 27623999466360000, 86663527737600000, 178424910048000000, 251893990656000000, 246954892800000000, 166020096000000000, 73244160000000000, 19148800000000000, 2252800000000000;

The row polynomials satisfy (n+1)*P(n,x) = (17*n+34+(20*n+40)*x)*P(n-1,x) for n >= 1, with P(0,x) = 680 + 800*x.

P[0, x_] := 680 + 800*x;
P[n_, x_] := P[n, x] = Expand[Together[((17*n + 34 + (20*n + 40)*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] = 680 + 800*x; for(n = 1, N, my(i = n + 1); v[i] = ((17*n + 34 + (20*n + 40)*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)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
        • 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

Related ?

Same fingerprint ?

• A038331 → basis A013611
• A038331 → 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.