A038226 → basis A007318 → basis A013609

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Submission text ?

Triangle begins:
  0:             361,              608,              256;
  1:            6859,            17328,            14592,              4096;
  2:          130321,           438976,           554496,            311296,             65536;
  3:         2476099,         10425680,         17559040,          14786560,           6225920,           1048576;
  4:        47045881,        237705504,        500432640,         561889280,         354877440,         119537664,          16777216;
  5:       893871739,       5269138672,      13311508224,       18682818560,       15732899840,        7949254656,        2231369728,         268435456;
  6:     16983563041,     114415582592,     337224875008,      567957684224,      597850193920,      402762235904,      169584099328,       40802189312,       4294967296;
  7:    322687697779,    2445633077904,    8237921946624,    16186794000384,    20446476632064,    17218085584896,     9666293661696,     3488587186176,     734439407616,      68719476736;
  8:   6131066257801,   51630031644640,  195650646232320,   439355837153280,   647471760015360,   654287252226048,   459148948930560,   220943855124480,   69771743723520,   13056700579840,   1099511627776;
  9: 116490258898219, 1079067661372976, 4543442784728320, 11478171245629440, 19331656834744320, 22791005952540672, 19192426065297408, 11544316430254080, 4860764812738560, 1364425210593280, 229797930205184, 17592186044416;

The row polynomials satisfy P(n,x) = (19+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 608*x + 256*x^2.

P[0, x_] := 361 + 608*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 361 + 608*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 16*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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • 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
            • 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
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A038322 → basis A013609
• A038322 → basis A013609 → D
• A038322 → basis A013609 → reverse
• A027467 → basis A013610 → x ↦ x/(x+1)

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.