A038331 → basis A013609 → reverse

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Submission text ?

Triangle begins:
  0:          100,           340,            289;
  1:         1000,          5100,           8670,           4913;
  2:        10000,         68000,         173400,         196520,           83521;
  3:       100000,        850000,        2890000,        4913000,         4176050,         1419857;
  4:      1000000,      10200000,       43350000,       98260000,       125281500,        85191420,         24137569;
  5:     10000000,     119000000,      606900000,     1719550000,      2923235000,      2981699700,       1689629830,        410338673;
  6:    100000000,    1360000000,     8092000000,    27512800000,     58464700000,     79511992000,      67585193200,      32827093840,       6975757441;
  7:   1000000000,   15300000000,   104040000000,   412692000000,   1052364600000,   1789019820000,    2027555796000,    1477219222800,     627818169690,    118587876497;
  8:  10000000000,  170000000000,  1300500000000,  5895600000000,  17539410000000,  35780396400000,   50688894900000,   49240640760000,   31390908484500,  11858787649700,   2015993900449;
  9: 100000000000, 1870000000000, 15895000000000, 81064500000000, 275619300000000, 655973934000000, 1115155687800000, 1354117620900000, 1150999977765000, 652233320733500, 221759329049390, 34271896307633;

The row polynomials satisfy P(n,x) = (10+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 340*x + 289*x^2.

P[0, x_] := 100 + 340*x + 289*x^2;
P[n_, x_] := P[n, x] = Expand[(10 + 17*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 100 + 340*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (10 + 17*x)*v[i - 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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013609 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 ?

• A038271 → basis A007318 → basis A013609
• A038331 → basis A013609
• A038331 → basis A013609 → D

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.