A225466 → D+1

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Submission text ?

Triangle begins:
   1:       1;
   2:       5,         3;
   3:      25,        39,          9;
   4:     125,       387,        216,         27;
   5:     625,      3471,       3537,       1026,         81;
   6:    3125,     29643,      49320,      24975,       4455,       243;
   7:   15625,    246519,     631449,     497610,     150660,     18225,       729;
   8:   78125,   2019027,    7685496,    8860887,    4054050,    816480,     71442,     2187;
   9:  390625,  16386591,   90597537,  147108906,   95501511,  28491750,   4092606,   271188,    6561;
  10: 1953125, 132264603, 1045732680, 2331317295, 2064852405, 856339533, 179605188, 19328706, 1003833, 19683;

The row polynomials satisfy P(n,x) = (5+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (5 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >=…?
    • 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

Related ?

Same fingerprint ?

• A038338 → basis A013610 → basis A008277
• A085880 → basis A013610 → basis A008277
• A196347 → basis A013610 → basis A008277
• A178822 → basis A013610 → basis A008277
• A038221 → basis A013610 → basis A008277
• A178820 → basis A013610 → basis A008277
• A156992 → basis A013610 → basis A008277
• A178821 → basis A013610 → basis A008277
• A038208 → basis A013610 → basis A008277
• A028326 → basis A013610 → basis A008277
• A013620 → basis A008277

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.