A038281 → θ+1

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Submission text ?

Triangle begins:
  0:         64,          96,           27;
  1:        512,        1152,          648,          108;
  2:       4096,       12288,        10368,         3456,          405;
  3:      32768,      122880,       138240,        69120,        16200,         1458;
  4:     262144,     1179648,      1658880,      1105920,       388800,        69984,        5103;
  5:    2097152,    11010048,     18579456,     15482880,      7257600,      1959552,      285768,       17496;
  6:   16777216,   100663296,    198180864,    198180864,    116121600,     41803776,     9144576,     1119744,      59049;
  7:  134217728,   905969664,   2038431744,   2378170368,   1672151040,    752467968,   219469824,    40310784,    4251528,    196830;
  8: 1073741824,  8053063680,  20384317440,  27179089920,  22295347200,  12039487488,  4389396480,  1074954240,  170061120,  15746400,   649539;
  9: 8589934592, 70866960384, 199313326080, 298969989120, 280284364800, 176579149824, 77253378048, 23648993280, 4988459520, 692841600, 57159432, 2125764;

The row polynomials satisfy (8+(3*n+6)*x)*P(n,x) = (64+(24*n+96)*x+(9*n+27)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 96*x + 27*x^2.

P[0, x_] := 64 + 96*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((64 + (24*n + 96)*x + (9*n + 27)*x^2)*P[n - 1, x])/(8 + (3*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 64 + 96*x + 27*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((64 + (24*n + 96)*x + (9*n + 27)*x^2)*v[i - 1])/(8 + (3*n + 6)*x)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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) [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) (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)

Related ?

Same fingerprint ?

• A027466 → basis A013610 → θ+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.