A038333 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:              798,               882;
  1:            22743,             50274,             27783;
  2:           576156,           1910412,           2111508,             777924;
  3:         13683705,          60496380,         100296630,           73902780,           20420505;
  4:        311988474,        1724146830,        3811271940,         4212458460,         2327937570,          514596726;
  5:       6915744507,       45862305678,      126724792005,       186752325060,       154807848405,        68441364558,        12607619787;
  6:     150170452152,     1161845077176,     3852433676952,      7096588352280,      7843597652520,      5201543706408,      1916358207624,       302582874888;
  7:    3209893414749,    28382215456728,   109794359793132,    242703321647976,    335313799645230,    296487991265256,    163848626751852,     51741671605848,      7148520419229;
  8:   67764416533590,   674077617097290,  2980132622956440,   7685605185519240,  12741924386518740,  14083179585099660,  10377079694283960,   4915458802555560,   1358218879653510,   166798809782010;
  9: 1416276305552031, 15653580219259290, 77855964774736995, 229470211967645880, 443843699463736110, 588676906657165788, 542202414026336910, 342443629911370680, 141933872923791795, 34860951244440090, 3853052505964431;

The row polynomials satisfy (n+1)*P(n,x) = (19*n+38+(21*n+42)*x)*P(n-1,x) for n >= 1, with P(0,x) = 798 + 882*x.

P[0, x_] := 798 + 882*x;
P[n_, x_] := P[n, x] = Expand[Together[((19*n + 38 + (21*n + 42)*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] = 798 + 882*x; for(n = 1, N, my(i = n + 1); v[i] = ((19*n + 38 + (21*n + 42)*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_A013610 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 A013610 → reverse
• A038333 → basis A013610

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.