A038328 → basis A013611 → basis A007318

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Submission text ?

Triangle begins:
  0:             484,              352,               64;
  1:           10648,            11616,             4224,              512;
  2:          234256,           340736,           185856,            45056,             4096;
  3:         5153632,          9370240,          6814720,          2478080,           450560,            32768;
  4:       113379904,        247374336,        224885760,        109035520,         29736960,          4325376,          262144;
  5:      2494357888,       6349274624,       6926481408,       4197867520,       1526497280,        333053952,        40370176,         2097152;
  6:     54875873536,     159638904832,     203176787968,     147764936704,      67165880320,      19539165184,      3552575488,       369098752,       16777216;
  7:   1207269217792,    3951062894592,    5747000573952,    4876242911232,    2659768860672,     967188676608,    234469982208,     36540776448,     3321888768,     134217728;
  8:  26559922791424,   96581537423360,  158042515783680,  153253348638720,   97524858224640,   42556301770752,  12895849021440,   2679656939520,   365407764480,   29527900160,   1073741824;
  9: 584318301411328, 2337273205645312, 4249587646627840, 4635913796321280, 3371573670051840, 1716437504753664, 624159092637696, 162119244840960, 29476226334720, 3572875919360, 259845521408, 8589934592;

The row polynomials satisfy P(n,x) = (22+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 352*x + 64*x^2.

P[0, x_] := 484 + 352*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 484 + 352*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 8*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 (−∞, -1]
  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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A013618 → basis A013610
• A013618 → basis A013610 → D
• A038303 → basis A013611 → reverse
• A013618 → basis A013610 → reverse
• A038303 → basis A013611 → D
• A038303 → basis A013611

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.