A038271 → D

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Submission text ?

Triangle begins:
  0:         0;
  1:         5;
  2:        70,         50;
  3:       735,       1050,        375;
  4:      6860,      14700,      10500,       2500;
  5:     60025,     171500,     183750,      87500,      15625;
  6:    504210,    1800750,    2572500,    1837500,     656250,      93750;
  7:   4117715,   17647350,   31513125,   30012500,   16078125,    4593750,    546875;
  8:  32941720,  164708600,  352947000,  420175000,  300125000,  128625000,  30625000,   3125000;
  9: 259416045, 1482377400, 3705943500, 5294205000, 4726968750, 2701125000, 964687500, 196875000, 17578125;

The row polynomials satisfy (n-1)*P(n,x) = (7*n+5*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.

P[0, x_] := 0;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[((7*n + 5*n*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] = 0; v[2] = 5; for(n = 2, N, my(i = n + 1); v[i] = ((7*n + 5*n*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 Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^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

Related ?

Same fingerprint ?

• A038271 → reverse
• A038329 → basis A013609 → x ↦ x/(x+1)
• A038334 → x ↦ x+1 → x ↦ x/(x+1)
• A038291 → basis A013611 → x ↦ x/(x+1)
• A038271: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
• A013612 → basis A013610 → basis A007318
• A038220 → basis A007318 → x ↦ x/(x+1)
• A013622 → basis A013611 → basis A007318

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.